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<div class="add">

<span class="kw">thus </span><a NAME="E1:100"/>
 <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10) or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_1"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span></span><br/>

<a NAME="E2:100_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10)
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:100_1"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">and </span><br/><a NAME="E5:100_1"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 10 )
 ;<br/>
<a NAME="E6:100_1"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
<font color="Maroon" title="c2">k</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E4:100_1"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="newton.html#D6" onmouseover="rs('newton/D6')" onmouseout="rh()">NEWTON:def 6</a></span>;<br/>
<a NAME="E7:100_1"/>
 not  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 10
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:100_1"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:100_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100_1"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E6:100_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="enumset1.html#D2" onmouseover="rs('enumset1/D2')" onmouseout="rh()">ENUMSET1:def 2</a>, <a class="ref" href="xprimes0.html#T4" onmouseover="rs('xprimes0/T4')" onmouseout="rh()">XPRIMES0:4</a>, <a class="ref" href="xprimes0.html#T6" onmouseover="rs('xprimes0/T6')" onmouseout="rh()">XPRIMES0:6</a>, <a class="ref" href="xprimes0.html#T8" onmouseover="rs('xprimes0/T8')" onmouseout="rh()">XPRIMES0:8</a>, <a class="ref" href="xprimes0.html#T9" onmouseover="rs('xprimes0/T9')" onmouseout="rh()">XPRIMES0:9</a>, <a class="ref" href="xprimes0.html#T10" onmouseover="rs('xprimes0/T10')" onmouseout="rh()">XPRIMES0:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10) )</span><br/>

<span class="kw">assume </span><a NAME="E2:100"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K9" title="DOMAIN_1:func.9">{</a><span class="default">2,3,5,7</span><a href="domain_1.html#K9" title="DOMAIN_1:func.9">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10)</span><br/>

<a NAME="E3:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A5</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" onmouseover="rs('enumset1/D2')" onmouseout="rh()">ENUMSET1:def 2</a></span>;<br/>
<a NAME="E4:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D6" onmouseover="rs('newton/D6')" onmouseout="rh()">NEWTON:def 6</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a>, <a class="ref" href="xprimes1.html#T5" onmouseover="rs('xprimes1/T5')" onmouseout="rh()">XPRIMES1:5</a>, <a class="ref" href="xprimes1.html#T7" onmouseover="rs('xprimes1/T7')" onmouseout="rh()">XPRIMES1:7</a></span>;<br/>
<a NAME="E5:100"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E67">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:100"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,10)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
