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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Olive" title="b2">a</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Olive" title="b2">a</font> holds <br/> <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b2">a</font></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Olive" title="b1">a</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Olive" title="b1">a</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Maroon" title="c3">a</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c3">a</font> implies  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> )</span><br/>





<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Maroon" title="c3">a</font> is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c3">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">xx</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">xx</font> &amp; <font color="Maroon" title="c3">a</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">xx</font> )
 ;<br/>
<b>set </b><font color="Maroon" title="c5">nx</font> =  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font>;<br/>
<b>set </b><font color="Maroon" title="c6">na</font> =  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font>;<br/>
<a NAME="E4:36"/>
( <font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a>, <a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E5:36"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1"/>
(  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="#E5:36"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>suppose </b></a><a NAME="E1:36_1_1"/><i><font color="Green" title="E31">B0</font></i>: 
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><a NAME="E2:36_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a>, <a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E3:36_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bcialg_6.html#T4" target="_self" title="BCIALG_6:th.4">Th3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">aa</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:36_1_1"/><i><font color="Green" title="E32">B1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">aa</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c7">aa</font> )
 ;<br/><a NAME="E6:36_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">ab</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:36_1_1"/><i><font color="Green" title="E33">B3</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ab</font> &amp; <font color="Maroon" title="c8">ab</font> is <a href="bcialg_1.html#V10" title="BCIALG_1:attr.10">atom</a> )
 ;<br/><a NAME="E9:36_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E5:36_1_1"><i><font color="Green" title="E32">B1</font></i></a>, <a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/><a NAME="E10:36_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E8:36_1_1"><i><font color="Green" title="E33">B3</font></i></a>, <a class="ref" href="bcialg_1.html#D14" title="BCIALG_1:def.14">BCIALG_1:def 14</a></i>;<br/><a NAME="E11:36_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:36"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/><a NAME="E12:36_1_1"/><b>then </b><i><font color="Green" title="E34">B7</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="bcialg_6.html#T4" target="_self" title="BCIALG_6:th.4">Th3</a></i>;<br/><a NAME="E13:36_1_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E14:36_1_1"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="#E1:36_1_1"><i><font color="Green" title="E31">B0</font></i></a>, <a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a>, <a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E12:36_1_1"><i><font color="Green" title="E34">B7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><a NAME="E1:36_1_2"/><i><font color="Green" title="E31">C1</font></i>: 
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span><br/><div class="add"><a NAME="E2:36_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a>, <a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">aa</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:36_1_2"/><i><font color="Green" title="E32">B1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">aa</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c7">aa</font> )
 ;<br/><a NAME="E5:36_1_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E6:36_1_2"/><b>then </b><i><font color="Green" title="E33">B2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
 <b>by </b><i><a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="bcialg_1.html#T37" title="BCIALG_1:th.37">BCIALG_1:37</a></i>;<br/><a NAME="E7:36_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p2">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E4:36_1_2"><i><font color="Green" title="E32">B1</font></i></a>, <a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/><a NAME="E8:36_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E6:36_1_2"><i><font color="Green" title="E33">B2</font></i></a>, <a class="ref" href="bcialg_6.html#T17" target="_self" title="BCIALG_6:th.17">Th17</a></i>;<br/><a NAME="E9:36_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p2">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="bcialg_6.html#T18" target="_self" title="BCIALG_6:th.18">Th17b</a></i>;<br/><a NAME="E10:36_1_2"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/><a NAME="E11:36_1_2"/><b>then </b><i><font color="Green" title="E34">B7</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E1:36_1_2"><i><font color="Green" title="E31">C1</font></i></a></i>;<br/><a NAME="E12:36_1_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> )</span><br/>

<b>assume </b><a NAME="E1:36_1_2_1"/><i><font color="Green" title="E35">C1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">xx</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:36_1_2_1"/><i><font color="Green" title="E36">C3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xx</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c9">xx</font> )
 ;<br/>
<a NAME="E4:36_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E3:36_1_2_1"><i><font color="Green" title="E36">C3</font></i></a>, <a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E5:36_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E6:36_1_2"><i><font color="Green" title="E33">B2</font></i></a>, <a class="ref" href="bcialg_6.html#T18" target="_self" title="BCIALG_6:th.18">Th17b</a></i>;<br/>
<a NAME="E6:36_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="bcialg_6.html#T17" target="_self" title="BCIALG_6:th.17">Th17</a></i>;<br/>
<a NAME="E7:36_1_2_1"/><b>then </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<a NAME="E8:36_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="bcialg_6.html#K3" title="BCIALG_6:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<b>hence </b><a NAME="E9:36_1_2_1"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a>, <a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E1:36_1_2_1"><i><font color="Green" title="E35">C1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:36_1_2"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="bcialg_6.html#D5" target="_self" title="BCIALG_6:def.5">Def8a</a>, <a class="txt" href="#E1:36"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E11:36_1_2"><i><font color="Green" title="E34">B7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
