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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bcialg_4.html#NM1" title="BCIALG_4:NM.1">BCI-Algebra_with_Condition(S)</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>  ex <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_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> st <br/>for <font color="Olive" title="b4">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> holds  <font color="Olive" title="b4">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b3">a</font></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">  ex <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_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> st <br/>for <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds  <font color="Olive" title="b2">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font></span><br/>



<a NAME="E1:16"/>
 ex <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_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> st <br/>for <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds  <font color="Olive" title="b2">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">t</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> holds <br/><font color="Olive" title="b1">t</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font> ) )
 
<b>by </b><i><a class="txt" href="bcialg_4.html#E8"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E2:16_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font><b> such that </b><br/><a NAME="E4:16_1"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_4.html#K1" title="BCIALG_4:func.1">*</a> <font color="Maroon" title="c3">y</font>
 ;<br/>
<a NAME="E5:16_1"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">t1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds  <font color="Olive" title="b1">t1</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">t1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">t1</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E1:16_1_1"/>
<font color="Maroon" title="c5">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</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">t</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">t2</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="E3:16_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c6">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t1</font> &amp; <font color="Maroon" title="c6">t2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c2">x</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 ;<br/>
<b>thus </b><a NAME="E4:16_1_1"/>
<font color="Maroon" title="c5">t1</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E4:16_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="bcialg_4.html#E3:16_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="bcialg_4.html#E8"><i><font color="Green" title="E3">Lm2</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/>


</div><b>end;</b></div>
<b>take </b>
<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"> for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds  <font color="Olive" title="b1">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font></span><br/>

<b>thus </b><a NAME="E6:16_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds  <font color="Olive" title="b1">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="bcialg_4.html#E5:16_1"><i><font color="Green" title="E6">A2</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
 ex <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_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> st <br/>for <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_4.html#K3" title="BCIALG_4:func.3">Condition_S</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds  <font color="Olive" title="b2">z</font> <a href="bcialg_1.html#R1" title="BCIALG_1:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
