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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>, <font color="Olive" title="b2">y</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="b3">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> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</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="b3">a</font> &amp; <font color="Olive" title="b2">y</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="b3">a</font> holds <br/> <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#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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> &amp; <font color="Maroon" title="c3">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>let </b><font color="Maroon" title="c4">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> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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="c4">a</font> &amp; <font color="Maroon" title="c3">y</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="c4">a</font> implies  <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#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>







<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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="c4">a</font> &amp; <font color="Maroon" title="c3">y</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="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E2:40"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</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_1.html#T40" title="BCIALG_1:th.40">BCIALG_1:40</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <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#T4" target="_self" title="BCIALG_6:th.4">Th3</a></i>;<br/>
<a NAME="E4:40"/>
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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <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="E5:40"/>
 <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#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <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="#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E1:40"><i><font color="Green" title="E29">A1</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/>


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