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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Olive">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">m</font>] ) </span>}</span>  &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Olive">r</font></span><a href="int_1.html#K1">/]</a></span> holds <br/> <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Olive">r</font></span><a href="int_1.html#K1">/]</a></span>;<br/>
<a NAME="E1:70_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:70_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E2:70_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<b>assume </b><a NAME="E3:70_1_1"/><i><font color="Green">E40</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/>

<a NAME="E4:70_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E5:70_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_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="E1:70_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:70_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">m</font>] )
 <b>by </b><i>, </i>;<br/><a NAME="E4:70_1_1_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E5:70_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:70_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:70_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/><b>assume </b><a NAME="E2:70_1_2_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><b>assume </b><a NAME="E3:70_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><b>set </b><font color="Maroon">r1</font> = <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1;<br/><i><font color="Green">E58</font></i>: <font color="Maroon">n</font> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E10:70_1_2_1"><i><font color="Green">E69</font></i></a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span><a href="int_1.html#K1">/]</a></span>

;<br/><a NAME="E5:70_1_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><b>set </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> ;<br/><b>set </b><font color="Maroon">X2</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_2"><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="E1:70_1_2_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:70_1_2_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">m</font>] )
 <b>by </b><i>, <a class="txt" href="nat_4.html#E14:70_1_2_1"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E4:70_1_2_1_2"/>
( ( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">m</font>] ) or ( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">m</font>] ) )
 <b>by </b><i/>;<br/><a NAME="E5:70_1_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  )
 ;<br/><b>hence </b><a NAME="E6:70_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E7:70_1_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_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="E1:70_1_2_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_3_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:70_1_2_1_3_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">m</font>] )
 <b>by </b><i/>;<br/><a NAME="E4:70_1_2_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="binop_2.html#K9">+</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:70_1_2_1_3_1_1"/><b>then </b>
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">m</font>] )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:70_1_2_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:70_1_2_1_3_1_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:70_1_2_1_3_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] )
 ;<br/><b>hence </b><a NAME="E3:70_1_2_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:70_1_2_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E10:70_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="nat_4.html#E14:70_1_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E11:70_1_2_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E13:70_1_2_1"/><i><font color="Green">E71</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">X1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E11:70_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E13:70_1_2_1"><i><font color="Green">E71</font></i></a>, </i>;<br/><a NAME="E14:70_1_2_1"/><i><font color="Green">E72</font></i>: 
0 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_4"/>
( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  or  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_4_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_4_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:70_1_2_1_4_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>hence </b><a NAME="E3:70_1_2_1_4_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E11:70_1_2_1"><i><font color="Green">E70</font></i></a>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="txt" href="nat_4.html#E10:70_1_2_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_4_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_4_2"/><i><font color="Green">E73</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_2_1_4_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span> 
 <b>by </b><i/>;<br/><div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_2_1_4_2_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_2_1_4_2_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:70_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E3:70_1_2_1_4_2_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zfmisc_1.html#T41">ZFMISC_1:41</a></i>;<br/><b>consider </b><font color="Maroon">p1</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E5:70_1_2_1_4_2_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p1</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p1</font>,<font color="Maroon">m</font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E5:70_1_2_1_4_2_1"><i><font color="Green">E78</font></i></a></i>;<br/><b>consider </b><font color="Maroon">p2</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E7:70_1_2_1_4_2_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">p2</font>,<font color="Maroon">m</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p2</font>,<font color="Maroon">m</font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E4:70_1"><i><font color="Green">E81</font></i></a></i>;<br/><a NAME="E8:70_1_2_1_4_2_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a>, <a class="txt" href="nat_4.html#E5:70_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>hence </b><a NAME="E9:70_1_2_1_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E5:70_1"><i><font color="Green">E82</font></i></a>, , <a class="txt" href="nat_4.html#E4:70_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_4_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_2_1_4_2_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_4_2_2_1"><b>suppose </b></a><a NAME="E1:70_1_2_1_4_2_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><div class="add"><a NAME="E2:70_1_2_1_4_2_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">X1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E1:70_1_2_1_4_2"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/><b>hence </b><a NAME="E3:70_1_2_1_4_2_2_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E10:70_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nat_4.html#E11:70_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E4:70_1_2_1_4_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2_1_4_2_2_2"><b>suppose </b></a><a NAME="E1:70_1_2_1_4_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><div class="add"><a NAME="E2:70_1_2_1_4_2_2_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a>, </i>;<br/><a NAME="E3:70_1_2_1_4_2_2_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E11:70_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E4:70_1_2_1_4_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="nat_4.html#E1:70_1_2_1_4_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E4:70_1_2_1_4_2_2_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E10:70_1_2_1"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:70_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E4:70_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>1</sub></font>(<font color="Olive">p</font>,<font color="Maroon">m</font>) where <font color="Olive">p</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">m</font>] ) </span>}</span>  &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>

<a NAME="E5:70_1"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a>, </i>;<br/>
<b>thus </b><a NAME="E6:70_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, , </i>;<br/>


</div>
