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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">a</font>, <font color="Olive" title="b3">b</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="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="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Olive" title="b3">b</font></span><br/>

<b>set </b><font color="Maroon" title="c2">P</font> =  <a href="bcialg_1.html#K5" title="BCIALG_1:func.5">AtomSet</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> is    <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="c5">b</font> implies <font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<b>set </b><font color="Maroon" title="c6">B</font> =  <a href="bcialg_1.html#K7" title="BCIALG_1:func.7">BranchV</a> <font color="Maroon" title="c5">b</font>;<br/>
<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E2">A0</font></i>: 
<font color="Maroon" title="c3">x</font> is    <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="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:7"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c3">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="c5">b</font>
 
<b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E2">A0</font></i></a></i>;<br/>
<a NAME="E3:7"/>
<font color="Maroon" title="c3">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> : <font color="Olive" title="b1">x1</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="c7">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="E5:7"/><i><font color="Green" title="E4">B1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c7">x1</font> is <a href="bcialg_1.html#V10" title="BCIALG_1:attr.10">atom</a> )
 ;<br/>
<b>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="E7:7"/><i><font color="Green" title="E5">C1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x3</font> &amp; <font color="Maroon" title="c5">b</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c8">x3</font> )
 <b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E8:7"/>
<font color="Maroon" title="c5">b</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">x</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="#E7:7"><i><font color="Green" title="E5">C1</font></i></a>, <a class="ref" href="bcialg_1.html#D11" title="BCIALG_1:def.11">BCIALG_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E9:7"/>
<font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="bcialg_1.html#K6" title="BCIALG_1:func.6">\</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="#E5:7"><i><font color="Green" title="E4">B1</font></i></a>, <a class="ref" href="bcialg_1.html#D14" title="BCIALG_1:def.14">BCIALG_1:def 14</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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