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<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">d</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E2:19"/>
<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">d</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">d</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">d</font> )
 ;<br/>
<b>consider </b><font color="Maroon">d1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E6:19_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">d2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:19_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E9:19_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:19_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E11:19_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a>, , </i>;<br/>
<a NAME="E12:19_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d2</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E13:19_1"/>
<font color="Maroon">d1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:19_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">d1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E15:19_1"/>
<font color="Maroon">d1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:19_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> ) </span>}</span> 
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:19"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:19"/>
<font color="Maroon">d1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E8:19"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">d1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>consider </b><font color="Maroon">d2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:19"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E11:19"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d2</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:19"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<a NAME="E13:19"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">d1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E14:19"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:19"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">d</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div>
