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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) &amp; ( <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) implies  not  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 )</span><br/>



<a NAME="E1:55"/><span class="lab"><font color="Green" title="E41">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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7
 
<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">assume </span><a NAME="E2:55"/>
( ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) &amp; ( <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  not  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_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="E1:55_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><span class="kw">suppose </span></a><a NAME="E1:55_1_1"/>
( ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <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">  not  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:55_1_1"/>
 not  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</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="55_1_2"><span class="kw">suppose </span></a><a NAME="E1:55_1_2"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  not  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:55_1_2"/>
 not  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ...  &amp;  not <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
  <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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="55_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 6 &amp; <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E1:55_1_2_1"/>
<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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">7 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:55_1_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:55_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 6 &amp; <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T16" target="_self" onmouseover="rs('number02/T16')" onmouseout="rh()">Th16</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>
