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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  implies ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 <span class="kw">and </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</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="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:6"/>
( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_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="E1:6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><span class="kw">suppose </span></a><a NAME="E1:6_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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/><div class="add"><a NAME="E2:6_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) or ( <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/><a NAME="E3:6_1_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  or  not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E10">A2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><span class="kw">suppose </span></a><a NAME="E1:6_1_2"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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/><div class="add"><a NAME="E2:6_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) or ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/><a NAME="E3:6_1_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  or  not <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_2"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E10">A2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_3"><span class="kw">suppose </span></a><a NAME="E1:6_1_3"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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/><div class="add"><a NAME="E2:6_1_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp; <font color="Maroon" title="c2">b</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  ) or ( <font color="Maroon" title="c1">a</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/><a NAME="E3:6_1_3"/><span class="kw">then </span>
(  not <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  or  not <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_3"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E10">A2</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></div><span class="kw">end;</span></div>

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