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<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font>
 ;<br/><div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:4_1_1"/>
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/><a NAME="E2:4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_1"><b>case </b></a><a NAME="E1:4_1_2_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then </b><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="polynom7.html#T3">POLYNOM7:3</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i>, <a class="ref" href="polynom7.html#T3">POLYNOM7:3</a></i>
;<br/><b>hence </b><a NAME="E3:4_1_2_1_1"/>
 ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2_1_2"><b>case </b></a><a NAME="E1:4_1_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>set </b><font color="Maroon">b</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font> : verum </span>}</span> ;<br/><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_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/><b>assume </b><a NAME="E1:4_1_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font><b> such that </b><br/><a NAME="E3:4_1_2_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/><b>thus </b><a NAME="E4:4_1_2_1_2_1"/>
 ex <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_2"/><i><font color="Green">E44</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font> : verum </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font> : verum </span>}</span>  )
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font><b> such that </b><br/><a NAME="E3:4_1_2_1_2_2"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/><b>consider </b><font color="Maroon">k'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font><b> such that </b><br/><a NAME="E5:4_1_2_1_2_2"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><font color="Maroon">k</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">k'</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><b>hence </b><font color="Maroon">y1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">b</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">k</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font> : verum </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_2_1_2_3"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font><b> such that </b><br/><a NAME="E5:4_1_2_1_2_3"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><b>hence </b><a NAME="E7:4_1_2_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n'</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n'</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font> ;<br/><a NAME="E3:4_1_2_1_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/><b>hence </b><a NAME="E4:4_1_2_1_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E8:4_1_2_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n'</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_2_1_2_5"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font><b> such that </b><br/><a NAME="E5:4_1_2_1_2_5"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E7:4_1_2_1_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><a NAME="E11:4_1_2_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_6"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_2_1_2_6"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">k'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n'</font><b> such that </b><br/><a NAME="E5:4_1_2_1_2_6"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><i><font color="Green">E59</font></i>: <font color="Maroon">k</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">k'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><font color="Maroon">u</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k'</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E8:4_1_2_1_2_6"/>
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_7"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:4_1_2_1_2_7"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:4_1_2_1_2_7"/><i><font color="Green">E62</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_2_1_2_7_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:4_1_2_1_2_7_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:4_1_2_1_2_7_1"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><b>hence </b><a NAME="E4:4_1_2_1_2_7_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:4_1_2_1_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 ;<br/></div><b>end;</b></div><a NAME="E14:4_1_2_1_2"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E15:4_1_2_1_2"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1_2_8"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:4_1_2_1_2_8"/><i><font color="Green">E65</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>

;<br/></div><b>end;</b></div><a NAME="E18:4_1_2_1_2"/><b>then </b>
<font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T37">POLYNOM1:37</a></i>;<br/><b>hence </b><a NAME="E19:4_1_2_1_2"/>
 ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:4_1"/>
 ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4"/>
( <font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font> iff  ex <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Olive">b</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T54">POLYNOM1:54</a></i>;<br/>


</div>
