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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>  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="b1">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 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> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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"> ( <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="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</a> 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> <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> )</span><br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E29">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="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  is <a href="bcialg_6.html#V1" title="BCIALG_6:attr.1">finite-period</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="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> <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></span><br/>

<b>set </b><font color="Maroon" title="c3">n</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="c4">m</font> =  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <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>;<br/>
<a NAME="E2:39"/>
<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="c2">x</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="#E1:39"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">zz</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="E4:39"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c5">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">x</font></span>)</span> &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="c5">zz</font> )
 ;<br/>
<a NAME="E5:39"/>
<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="c2">x</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="#E4:39"><i><font color="Green" title="E30">A3</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="E6:39"/><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="c2">x</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="ref" href="bcialg_6.html#T18" target="_self" title="BCIALG_6:th.18">Th17b</a></i>;<br/>
<a NAME="E7:39"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</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_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>;<br/>
<a NAME="E8:39"/><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> <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="c2">x</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="E9:39"/><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="c2">x</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="c2">x</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="#E1:39"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<a NAME="E10:39"/><b>then </b><i><font color="Green" title="E31">C1</font></i>: 
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">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="#E1:39"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<a NAME="E11:39"/>
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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="#E1:39"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">zz</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="E13:39"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c6">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></span>)</span> &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="c6">zz</font> )
 ;<br/>
<a NAME="E14:39"/>
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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="#E13:39"><i><font color="Green" title="E32">A3</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="E15:39"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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_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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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="ref" href="bcialg_6.html#T18" target="_self" title="BCIALG_6:th.18">Th17b</a></i>;<br/>
<a NAME="E16:39"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_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="p2">(<span class="default"><a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></i>;<br/>
<a NAME="E17:39"/><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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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#T21" target="_self" title="BCIALG_6:th.21">Th19</a></i>;<br/>
<a NAME="E18:39"/><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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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="E19:39"/><b>then </b>
( <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span></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> <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="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="#E1:39"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<a NAME="E20:39"/><b>then </b>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">ord</a> <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>
 
<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="#E1:39"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:39"/>
 <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> <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>
 <b>by </b><i><a class="txt" href="#E10:39"><i><font color="Green" title="E31">C1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
