<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:7"/>
 <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR1" title="BCIALG_1:NR.1">is_B</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR2" title="BCIALG_1:NR.2">is_C</a>  &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR3" title="BCIALG_1:NR.3">is_I</a>  &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR5" title="BCIALG_1:NR.5">is_BCI-4</a>  &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_1.html#V8" title="BCIALG_1:attr.8">being_BCK-5</a> &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_4.html#V2" title="BCIALG_4:attr.2">with_condition_S</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_1.html#D3" title="BCIALG_1:def.3">BCIALG_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> </span><br/>

<b>thus </b><a NAME="E1:7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> 
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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><b>end;</b></div>

<b>thus </b><a NAME="E2:7"/>
 <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR2" title="BCIALG_1:NR.2">is_C</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR3" title="BCIALG_1:NR.3">is_I</a>  &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR5" title="BCIALG_1:NR.5">is_BCI-4</a>  &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_1.html#V8" title="BCIALG_1:attr.8">being_BCK-5</a> &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_4.html#V2" title="BCIALG_4:attr.2">with_condition_S</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_1.html#D4" title="BCIALG_1:def.4">BCIALG_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> </span><br/>

<b>thus </b><a NAME="E1:7_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> 
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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><b>end;</b></div>

<b>thus </b><a NAME="E3:7"/>
 <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR3" title="BCIALG_1:NR.3">is_I</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR5" title="BCIALG_1:NR.5">is_BCI-4</a>  &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_1.html#V8" title="BCIALG_1:attr.8">being_BCK-5</a> &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_4.html#V2" title="BCIALG_4:attr.2">with_condition_S</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c1">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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> </span><br/>

<b>thus </b><a NAME="E1:7_3"/>
<font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c1">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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> 
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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><b>end;</b></div>

<b>thus </b><a NAME="E4:7"/>
 <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  <a href="bcialg_1.html#NR5" title="BCIALG_1:NR.5">is_BCI-4</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_1.html#V8" title="BCIALG_1:attr.8">being_BCK-5</a> &amp;  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_4.html#V2" title="BCIALG_4:attr.2">with_condition_S</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  or  not <font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c1">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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )</span><br/>

<b>thus </b><a NAME="E1:7_4"/>
(  not <font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  or  not <font color="Maroon" title="c2">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c1">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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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><b>end;</b></div>

<b>thus </b><a NAME="E5:7"/>
 <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_1.html#V8" title="BCIALG_1:attr.8">being_BCK-5</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a>  is <a href="bcialg_4.html#V2" title="BCIALG_4:attr.2">with_condition_S</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_1.html#D8" title="BCIALG_1:def.8">BCIALG_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> </span><br/>

<b>thus </b><a NAME="E1:7_5"/>
<font color="Maroon" title="c1">x</font> <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> <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> 
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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><b>end;</b></div>

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="bcialg_4.html#K2" title="BCIALG_4:func.2">BCI_S-EXAMPLE</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bcialg_4.html#D2" target="_self" title="BCIALG_4:def.2">BCIALG_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<b>thus </b><a NAME="E6:7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</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>
