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

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">K</font> being   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">RK</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">K</font> holds   <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">RK</font>) is   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be    <a href="bcialg_1.html#M1" title="BCIALG_1:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">RK</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">K</font> holds   <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b2">RK</font>) is   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be   <a href="bcialg_1.html#V12" title="BCIALG_1:attr.12">closed</a>   <a href="bcialg_1.html#M2" title="BCIALG_1:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">RK</font> being    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">K</font> holds   <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Olive" title="b1">RK</font>) is   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a></span><br/><span class="kw">let </span><font color="Maroon" title="c4">RK</font> be    <a href="bcialg_2.html#M5" title="BCIALG_2:mode.5">I-congruence</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) is   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a></span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">IT</font> =  <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="bcialg_1.html#L2" title="BCIALG_1:struct.2">BCIStr_0</a>  ;<br/>
<a NAME="E2:80"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c5">IT</font> is  <a href="bcialg_1.html#V7" title="BCIALG_1:attr.7">being_BCI-4</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">IT</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bcialg_1.html#D7" onmouseover="rs('bcialg_1/D7')" onmouseout="rh()">BCIALG_1:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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> <font color="Maroon" title="c5">IT</font> or  not <font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">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="c5">IT</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x2</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c11">y2</font> = <font color="Maroon" title="c9">y1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">assume </span><a NAME="E3:80_1"/>
( <font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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> <font color="Maroon" title="c5">IT</font> &amp; <font color="Maroon" title="c7">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">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="c5">IT</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E4:80_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c11">y2</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> &amp; <font color="Maroon" title="c11">y2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">x2</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:80_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D7" onmouseover="rs('bcialg_1/D7')" onmouseout="rh()">BCIALG_1:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="80_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">IT</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">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="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">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> <font color="Maroon" title="c5">IT</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">IT</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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="c8">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c5">IT</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">x2</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c13">y2</font> = <font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c14">z2</font> = <font color="Maroon" title="c11">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:80_2"/>
( <font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font> &amp; <font color="Maroon" title="c14">z2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E4:80_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">z2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">z1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E5:80_2"/>
<font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E6:80_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</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="c14">z2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</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="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">z1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_2"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E7:80_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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="c8">z</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c5">IT</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D3" onmouseover="rs('bcialg_1/D3')" onmouseout="rh()">BCIALG_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:80"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">IT</font> is  <a href="bcialg_1.html#V4" title="BCIALG_1:attr.4">being_C</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="80_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">IT</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bcialg_1.html#D4" onmouseover="rs('bcialg_1/D4')" onmouseout="rh()">BCIALG_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">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="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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> <font color="Maroon" title="c5">IT</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x2</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c13">y2</font> = <font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c14">z2</font> = <font color="Maroon" title="c11">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:80_3"/>
<font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E4:80_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E5:80_3"/>
<font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E6:80_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E7:80_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</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="c12">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c14">z2</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c13">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_3"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:80_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">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="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">z</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">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> <font color="Maroon" title="c5">IT</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D4" onmouseover="rs('bcialg_1/D4')" onmouseout="rh()">BCIALG_1:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:80"/>
<font color="Maroon" title="c5">IT</font> is  <a href="bcialg_1.html#V5" title="BCIALG_1:attr.5">being_I</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="80_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">IT</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bcialg_1.html#D5" onmouseover="rs('bcialg_1/D5')" onmouseout="rh()">BCIALG_1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">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="c5">IT</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="bcialg_6.html#K6" title="BCIALG_6:func.6">Union</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c7">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:80_4"/>
<font color="Maroon" title="c8">x2</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">\</a> <font color="Maroon" title="c7">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D12" target="_self" onmouseover="rs('bcialg_6/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:80_4"/>
<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">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="c5">IT</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D5" onmouseover="rs('bcialg_1/D5')" onmouseout="rh()">BCIALG_1:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:80"/>
 <a href="bcialg_6.html#K9" title="BCIALG_6:func.9">HK</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font>) is   <a href="bcialg_1.html#NM1" title="BCIALG_1:NM.1">BCI-algebra</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="bcialg_1.html#D3" onmouseover="rs('bcialg_1/D3')" onmouseout="rh()">BCIALG_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
