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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">m</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">  for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> holds <br/><span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> <font color="Olive" title="b1">p</font></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>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">m</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</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"> ( <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> <font color="Maroon" title="c3">p</font></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>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">m</font>) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">A</font> =  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">m</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D</font> = <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> <font color="Maroon" title="c3">p</font></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>,<font color="Maroon" title="c2">m</font>)</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:115"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c3">p</font> <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"> <span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> <font color="Maroon" title="c3">p</font></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>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="number14.html#K8" title="NUMBER14:func.8">primeNumbers</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">m</font>)</span><br/>

<span class="kw">assume </span><a NAME="E2:115"/>
<span class="p1">(<span class="default"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> <font color="Maroon" title="c3">p</font></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>,<font color="Maroon" title="c2">m</font>)</span>)</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>,<font color="Maroon" title="c2">m</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"> 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="E4:115"/><span class="lab"><font color="Green" title="E80">A2</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"><a href="number14.html#K7" title="NUMBER14:func.7">multiples</a> <font color="Maroon" title="c3">p</font></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>,<font color="Maroon" title="c2">m</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:115"/><span class="lab"><font color="Green" title="E81">A3</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>,<font color="Maroon" title="c2">m</font>)
 <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/>
<a NAME="E6:115"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
<font color="Maroon" title="c6">x</font> is    <a href="number14.html#M3" title="NUMBER14:mode.3">Multiple</a> of <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="number14.html#T61" target="_self" onmouseover="rs('number14/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E7:115"/>
<font color="Maroon" title="c6">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> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</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="E9:115"/><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E10:115"/><span class="lab"><font color="Green" title="E84">A6</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E11:115"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font>
 ;<br/>
<a NAME="E12:115"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <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="#E1:115"><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/>
<a NAME="E13:115"/>
<font color="Maroon" title="c7">n</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="#E5:115"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E9:115"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="newton.html#D6" onmouseover="rs('newton/D6')" onmouseout="rh()">NEWTON:def 6</a></span>;<br/>
<a NAME="E14:115"/><span class="kw">then </span>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:115"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E9:115"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="number14.html#D15" target="_self" onmouseover="rs('number14/D15')" onmouseout="rh()">Def15</a></span>;<br/>
<a NAME="E15:115"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:115"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E12:115"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="E16:115"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</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/>


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