<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">A</font> =  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10);<br/>
<span class="kw">assume </span><a NAME="E1:117"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ; <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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:117_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T71" target="_self" onmouseover="rs('number14/T71')" onmouseout="rh()">Th71</a>, <a class="ref" href="card_2.html#T59" onmouseover="rs('card_2/T59')" onmouseout="rh()">CARD_2:59</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_1_2"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <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">  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c3">m</font> being    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 3<span class="kw"> such that </span><br/><a NAME="E3:117_1_2"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
<font color="Maroon" title="c3">m</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> (<font color="Maroon" title="c1">k</font>,10)
 <span class="kw">and </span><br/><a NAME="E4:117_1_2"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c3">m</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T84" target="_self" onmouseover="rs('number14/T84')" onmouseout="rh()">Th84</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">D</font> = <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span>;<br/><div><span class="lab"><font color="Green" title="E82">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_1_2_1"><span class="kw">now </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">  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:117_1_2_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c3">m</font> as    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T61" target="_self" onmouseover="rs('number14/T61')" onmouseout="rh()">Th61</a></span>;<br/><a NAME="E3:117_1_2_1"/>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#D15" target="_self" onmouseover="rs('number14/D15')" onmouseout="rh()">Def15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:117_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_2"><span class="lab"><font color="Green" title="E81">A3</font></span></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">set </span><font color="Maroon" title="c5">W</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E6:117_1_2"/>
10 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 5
 ;<br/><a NAME="E7:117_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T83" target="_self" onmouseover="rs('number14/T83')" onmouseout="rh()">Th83</a></span>;<br/><a NAME="E8:117_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A5</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_1_2"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="card_2.html#T41" onmouseover="rs('card_2/T41')" onmouseout="rh()">CARD_2:41</a></span>;<br/><a NAME="E9:117_1_2"/><span class="lab"><font color="Green" title="E84">A6</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 10
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T73" target="_self" onmouseover="rs('number14/T73')" onmouseout="rh()">Th73</a></span>;<br/><a NAME="E10:117_1_2"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117_1_2"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_1_2_2"><span class="kw">now </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">  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:117_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:117_1_2_2"/><span class="lab"><font color="Green" title="E86">A8</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:117_1_2_2"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_2_2"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="newton.html#D6" onmouseover="rs('newton/D6')" onmouseout="rh()">NEWTON:def 6</a></span>;<br/><a NAME="E6:117_1_2_2"/><span class="lab"><font color="Green" title="E88">A10</font></span>: 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_2_2"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="calcul_2.html#T1" onmouseover="rs('calcul_2/T1')" onmouseout="rh()">CALCUL_2:1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_1_2_2_1"/>
( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> or <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_2_2"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_1_2_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</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"> contradiction</span><br/><div class="add"><a NAME="E2:117_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T62" target="_self" onmouseover="rs('number14/T62')" onmouseout="rh()">Th62</a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:117_1_2"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E6:117_1_2_2"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_1_2_2_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> contradiction</span><br/><div class="add"><a NAME="E2:117_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:117_1_2_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:117_1_2"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E6:117_1_2_2"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="ref" href="number14.html#T60" target="_self" onmouseover="rs('number14/T60')" onmouseout="rh()">Th60</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E12:117_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A11</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_1_2"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="card_2.html#T40" onmouseover="rs('card_2/T40')" onmouseout="rh()">CARD_2:40</a></span>;<br/><a NAME="E13:117_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_2"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E14:117_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><a NAME="E15:117_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> 2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> (<font color="Maroon" title="c1">k</font>,10)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><a NAME="E16:117_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 10 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:117_1_2"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E12:117_1_2"><span class="lab"><font color="Green" title="E86">A11</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a>, <a class="ref" href="nat_1.html#T43" onmouseover="rs('nat_1/T43')" onmouseout="rh()">NAT_1:43</a></span>;<br/><span class="kw">hence </span><a NAME="E17:117_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,10)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 ; <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></div><span class="kw">end;</span></div>

</div>
