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<b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> holds  <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font>
 ;<br/>

<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="E2:23"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<a NAME="E3:23"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E5:23"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</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="E6:23"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_3.html#T55">NAT_3:55</a></i>;<br/>
<a NAME="E7:23"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E8:23"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:23"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#T56">NAT_3:56</a></i>;<br/>
<b>hence </b><a NAME="E10:23"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>


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