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<span class="kw">let </span><font color="Maroon" title="c17">k</font>, <font color="Maroon" title="c18">n</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"> ( <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span> is  <a href="number02.html#V1" title="NUMBER02:attr.1">composite</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104"/><span class="lab"><font color="Green" title="E83">A1</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:104"/><span class="lab"><font color="Green" title="E84">A2</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span> is  <a href="number02.html#V1" title="NUMBER02:attr.1">composite</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c19">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span>;<br/>
<a NAME="E3:104"/><span class="lab"><font color="Green" title="E85">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" onmouseover="rs('wsierp_1/T1')" onmouseout="rh()">WSIERP_1:1</a></span>;<br/>
<a NAME="E4:104"/><span class="lab"><font color="Green" title="E86">A4</font></span>: 
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="number07.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="newton.html#T9" onmouseover="rs('newton/T9')" onmouseout="rh()">NEWTON:9</a></span>;<br/>
<a NAME="E5:104"/>
<font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E6:104"/><span class="kw">then </span>
28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E7:104"/><span class="kw">then </span>
<span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 28 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<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>;<br/>
<a NAME="E8:104"/><span class="kw">then </span>
<span class="p1">(<span class="default">28 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:104"/><span class="kw">then </span>
4 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E84">A2</font></span></a>, <a class="ref" href="pepin.html#T66" onmouseover="rs('pepin/T66')" onmouseout="rh()">PEPIN:66</a></span>;<br/>
<a NAME="E10:104"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E86">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E11:104"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 5 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T96" onmouseover="rs('xreal_1/T96')" onmouseout="rh()">XREAL_1:96</a></span>;<br/>
<a NAME="E12:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 25 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E85">A3</font></span></a>, <a class="txt" href="number07.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:104"/>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span> is  <a href="number02.html#V1" title="NUMBER02:attr.1">composite</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"> contradiction</span><br/>

<a NAME="E14:104"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 2</span>)</span> 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="#E12:104"><span class="lab"><font color="Green" title="E87">A5</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="number02.html#D1" onmouseover="rs('number02/D1')" onmouseout="rh()">NUMBER02:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:104"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E84">A2</font></span></a>, <a class="txt" href="#E12:104"><span class="lab"><font color="Green" title="E87">A5</font></span></a>, <a class="ref" href="number07.html#T77" target="_self" onmouseover="rs('number07/T77')" onmouseout="rh()">Th77</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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