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<div class="add">

<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">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#K10" title="BCIALG_6:func.10">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/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K10" title="BCIALG_6:func.10">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/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K10" title="BCIALG_6:func.10">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/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">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/>







<b>reconsider </b><font color="Maroon" title="c5">IT</font> =  <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">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="E2:81"/><i><font color="Green" title="E44">C1</font></i>: 
<font color="Maroon" title="c5">IT</font> is <a href="bcialg_1.html#V3" title="BCIALG_1:attr.3">being_B</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><b>let </b><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" 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="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/><b>reconsider </b><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#K7" title="BCIALG_6:func.7">Union</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> ;<br/><b>reconsider </b><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="E3:81_1_1"/><i><font color="Green" title="E45">AA</font></i>: 
( <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#K11" title="BCIALG_6:func.11">\</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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font> &amp; <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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font> )
 <b>by </b><i><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/><a NAME="E4:81_1_1"/><b>then </b>
<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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">z1</font> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/><a NAME="E5:81_1_1"/><b>then </b>
<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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">z1</font> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:81_1_1"><i><font color="Green" title="E45">AA</font></i></a>, <a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/><b>hence </b><a NAME="E6:81_1_1"/>
<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>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D3" title="BCIALG_1:def.3">BCIALG_1:def 3</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="E2:81_1"/>
<font color="Maroon" title="c5">IT</font> is <a href="bcialg_1.html#V3" title="BCIALG_1:attr.3">being_B</a>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D3" title="BCIALG_1:def.3">BCIALG_1:def 3</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>
<a NAME="E3:81"/><i><font color="Green" title="E45">C3</font></i>: 
<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" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><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>; <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="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/>

<b>reconsider </b><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#K7" title="BCIALG_6:func.7">Union</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> ;<br/>
<b>reconsider </b><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="E3:81_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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font> &amp; <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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font> )
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/>
<a NAME="E4:81_2"/><b>then </b>
( <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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font> &amp; <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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font> )
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/>
<a NAME="E5:81_2"/><b>then </b>
<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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="bcialg_6.html#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:81_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> <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>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D4" title="BCIALG_1:def.4">BCIALG_1:def 4</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>
<a NAME="E4:81"/><i><font color="Green" title="E46">C5</font></i>: 
<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" onclick="hs2(this)" href="javascript:()" title="81_3"><b>proof </b></a><div class="add">

<b>let </b><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>; <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="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/>

<b>reconsider </b><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#K7" title="BCIALG_6:func.7">Union</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> ;<br/>
<b>reconsider </b><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="E3:81_3"/>
<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#K11" title="BCIALG_6:func.11">\</a> <font color="Maroon" title="c7">x1</font>
 
<b>by </b><i><a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:81_3"/>
<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>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</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>
<a NAME="E5:81"/>
<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" onclick="hs2(this)" href="javascript:()" title="81_4"><b>proof </b></a><div class="add">

<b>let </b><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>; <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="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/>

<b>assume </b><a NAME="E1:81_4"/><i><font color="Green" title="E47">D1</font></i>: 
( <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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>reconsider </b><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#K7" title="BCIALG_6:func.7">Union</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">RK</font> ;<br/>
<b>reconsider </b><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/>
<a NAME="E4:81_4"/>
( <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> )
 
<b>by </b><i><a class="txt" href="#E1:81_4"><i><font color="Green" title="E47">D1</font></i></a>, <a class="ref" href="bcialg_6.html#D12" target="_self" title="BCIALG_6:def.12">Def6</a></i>;<br/>
<b>hence </b><a NAME="E5:81_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="bcialg_1.html#D7" title="BCIALG_1:def.7">BCIALG_1:def 7</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="E6:81"/>
 <a href="bcialg_6.html#K10" title="BCIALG_6:func.10">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>
 <b>by </b><i><a class="txt" href="#E2:81"><i><font color="Green" title="E44">C1</font></i></a>, <a class="txt" href="#E3:81"><i><font color="Green" title="E45">C3</font></i></a>, <a class="txt" href="#E4:81"><i><font color="Green" title="E46">C5</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>
