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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>, <font color="Maroon" title="c4">r0</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) implies <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3 )</span><br/>







<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</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"> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_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="E1:40_1"/>
( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="number14.html#R5" title="NUMBER14:pred.5">give_three_different_remainders_upon_dividing_by</a> 3 or  not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="number14.html#R5" title="NUMBER14:pred.5">give_three_different_remainders_upon_dividing_by</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1"/><span class="lab"><font color="Green" title="E34">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="number14.html#R5" title="NUMBER14:pred.5">give_three_different_remainders_upon_dividing_by</a> 3
 ; <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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3</span><br/><div class="add"><a NAME="E2:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3,<font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3,<font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a> 
 ;<br/><a NAME="E3:40_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E1:40"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="number14.html#D10" target="_self" onmouseover="rs('number14/D10')" onmouseout="rh()">Def10</a></span>;<br/><span class="kw">assume </span><a NAME="E4:40_1_1"/>
( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number14.html#D8" target="_self" onmouseover="rs('number14/D8')" onmouseout="rh()">NUMBER14:def 8</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 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) or ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) or (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) )</span><br/><span class="kw">hence </span><a NAME="E5:40_1_1"/>
( (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) or ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) or (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E3:40_1_1"><span class="lab"><font color="Green" title="E36">A4</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="40_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="number14.html#R5" title="NUMBER14:pred.5">give_three_different_remainders_upon_dividing_by</a> 3
 ; <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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3</span><br/><div class="add"><a NAME="E2:40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A6</font></span>: 
 not <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3,<font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3,<font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a> 
 ;<br/><span class="kw">assume </span><a NAME="E3:40_1_2"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 not <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3
 ; <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="E4:40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
( ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) &amp; ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) &amp; (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) &amp; ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> or 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font> ) )
 ;<br/><a NAME="E5:40_1_2"/><span class="lab"><font color="Green" title="E38">A9</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="c4">r0</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r0</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/><a NAME="E6:40_1_2"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r0</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/><a NAME="E7:40_1_2"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">r0</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r0</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">a3</font> = <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 as    <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">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a>, <a class="ref" href="int_1.html#T57" onmouseover="rs('int_1/T57')" onmouseout="rh()">INT_1:57</a></span>;<br/><a NAME="E9:40_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="ref" href="int_1.html#T58" onmouseover="rs('int_1/T58')" onmouseout="rh()">INT_1:58</a></span>;<br/><a NAME="E10:40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 not  not <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c5">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_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="E1:40_1_2_1"/>
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 or <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 or ( <font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 &amp; <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 &amp; <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_1"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 ; <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="E2:40_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="number14.html#D10" target="_self" onmouseover="rs('number14/D10')" onmouseout="rh()">Def10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_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="E1:40_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_1_1_1"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:40_1_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E6:40_1_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E1:40_1_2_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E2:40_1_2_1_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="number14.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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="40_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_1_1_2"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><span class="kw">hence </span><a NAME="E2:40_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E1:40_1_2_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E2:40_1_2_1_1"><span class="lab"><font color="Green" title="E43">A14</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="40_1_2_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_2_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:40_1_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E6:40_1_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E1:40_1_2_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E2:40_1_2_1_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="number14.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="txt" href="number14.html#E6"><span class="lab"><font color="Green" title="E6">Lm6</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a>, <a class="ref" href="number02.html#T9" onmouseover="rs('number02/T9')" onmouseout="rh()">NUMBER02:9</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 ; <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="E2:40_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="number14.html#D10" target="_self" onmouseover="rs('number14/D10')" onmouseout="rh()">Def10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_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="E1:40_1_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_2_1_1"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:40_1_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E7:40_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E1:40_1_2_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E2:40_1_2_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="number14.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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="40_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_2_1_2"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><span class="kw">hence </span><a NAME="E2:40_1_2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E1:40_1_2_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E2:40_1_2_1_2"><span class="lab"><font color="Green" title="E43">A16</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="40_1_2_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_2_1_3"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_2_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:40_1_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E7:40_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E1:40_1_2_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E2:40_1_2_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="number14.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="txt" href="number14.html#E6"><span class="lab"><font color="Green" title="E6">Lm6</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a>, <a class="ref" href="number02.html#T9" onmouseover="rs('number02/T9')" onmouseout="rh()">NUMBER02:9</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:40_1_2_1_3"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 <span class="kw">and </span><br/><a NAME="E2:40_1_2_1_3"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 &amp; <font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 )
 ; <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"><span class="kw">reconsider </span><font color="Maroon" title="c6">b3</font> = <font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 as    <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">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a>, <a class="ref" href="int_1.html#T57" onmouseover="rs('int_1/T57')" onmouseout="rh()">INT_1:57</a></span>;<br/><a NAME="E4:40_1_2_1_3"/>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="ref" href="int_1.html#T58" onmouseover="rs('int_1/T58')" onmouseout="rh()">INT_1:58</a></span>;<br/><a NAME="E5:40_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">b3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E6:40_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A19</font></span>: 
 not  not <font color="Maroon" title="c6">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c6">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><a NAME="E7:40_1_2_1_3"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E2:40_1_2_1_3"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="number14.html#D10" target="_self" onmouseover="rs('number14/D10')" onmouseout="rh()">Def10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_3_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="E1:40_1_2_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_2_1_3"><span class="lab"><font color="Green" title="E44">A19</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_3_1_1"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_2_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E6:40_1_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E7:40_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E1:40_1_2_1_3"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E7:40_1_2_1_3"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="number14.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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="40_1_2_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_3_1_2"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><span class="kw">hence </span><a NAME="E2:40_1_2_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E1:40_1_2_1_3"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E7:40_1_2_1_3"><span class="lab"><font color="Green" title="E45">A20</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="40_1_2_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_3_1_3"/>
<font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_2_1_3_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E6:40_1_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E7:40_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E1:40_1_2_1_3"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E7:40_1_2_1_3"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="number14.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="txt" href="number14.html#E6"><span class="lab"><font color="Green" title="E6">Lm6</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a>, <a class="ref" href="number02.html#T9" onmouseover="rs('number02/T9')" onmouseout="rh()">NUMBER02:9</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
