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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>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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>;<br/>



<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E2:68"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">xx</font> where <font color="Olive" title="b1">xx</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">xx</font> is <a href="bcialg_1.html#V10" title="BCIALG_1:attr.10">atom</a> </span>}</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="E4:68"/><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">xx</font> &amp; <font color="Maroon" title="c4">xx</font> is <a href="bcialg_1.html#V10" title="BCIALG_1:attr.10">atom</a> )
 ;<br/>
<b>assume </b><a NAME="E5:68"/>
<span class="p1">(<span class="default"><a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:68"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c5">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c2">a</font> <b>by </b><i><a class="txt" href="bcialg_1.html#E7:68"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">z2</font> = <font color="Maroon" title="c5">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c3">b</font> <b>by </b><i><a class="txt" href="bcialg_1.html#E7:68"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:68"/>
<font color="Maroon" title="c6">z1</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">z2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bcialg_1.html#T39" target="_self" title="BCIALG_1:th.39">Th39</a></i>;<br/>
<a NAME="E11:68"/><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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x3</font> where <font color="Olive" title="b1">x3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">b</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">x3</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="bcialg_1.html#D5" target="_self" title="BCIALG_1:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x3</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="E13:68"/><i><font color="Green" title="E38">A4</font></i>: 
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x3</font> &amp; <font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">b</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c8">x3</font> )
 ;<br/>
<a NAME="E14:68"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.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="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="bcialg_1.html#E13:68"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="bcialg_1.html#D11" target="_self" title="BCIALG_1:def.11">Def11</a></i>;<br/>
<a NAME="E15:68"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">b</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="ref" href="bcialg_1.html#T2" target="_self" title="BCIALG_1:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E16:68"/>
contradiction
 <b>by </b><i><a class="txt" href="bcialg_1.html#E1:68"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="bcialg_1.html#E4:68"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="bcialg_1.html#D14" target="_self" title="BCIALG_1:def.14">Def14</a></i>;<br/>


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