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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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</a> <font color="Maroon" title="c1">X</font> holds  <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> <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></span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K4" title="BCIALG_1:func.4">BCK-part</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> <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></span><br/>



<a NAME="E1:37"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x1</font> where <font color="Olive" title="b1">x1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> :  <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="Olive" title="b1">x1</font> </span>}</span>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> :  <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="Olive" title="b1">y1</font> </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">x1</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="E3:37"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</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="c4">x1</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">y1</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="E5:37"/><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</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="c5">y1</font> )
 <b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E6:37"/>
<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_1.html#K2" title="BCIALG_1:func.2">`</a>  <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_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T9" target="_self" title="BCIALG_1:th.9">Th9</a></i>;<br/>
<a NAME="E7:37"/><b>then </b>
<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_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</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> 
 
<b>by </b><i><a class="txt" href="#E3:37"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="bcialg_1.html#D11" target="_self" title="BCIALG_1:def.11">Def11</a></i>;<br/>
<a NAME="E8:37"/><b>then </b>
<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_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> 
 
<b>by </b><i><a class="txt" href="#E5:37"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="bcialg_1.html#D11" target="_self" title="BCIALG_1:def.11">Def11</a></i>;<br/>
<a NAME="E9:37"/><b>then </b>
<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_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" target="_self" title="BCIALG_1:def.5">Def5</a></i>;<br/>
<a NAME="E10:37"/><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_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D11" target="_self" title="BCIALG_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E11:37"/>
<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>
 ; <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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