<?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">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#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Olive" title="b2">RK</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="c2">K</font> holds   <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Olive" title="b1">RK</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c2">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#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">K</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">h</font> =  <a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c3">RK</font>;<br/>
<span class="kw">thus </span><a NAME="E1:69"/>
 <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">K</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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="c2">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</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"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c3">RK</font></span>)</span>
 ;<br/>
<a NAME="E5:69_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">RK</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">RK</font>,<font color="Maroon" title="c7">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="bcialg_6.html#D10" target="_self" onmouseover="rs('bcialg_6/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E6:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">RK</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E7:69_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<a NAME="E8:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D12" onmouseover="rs('bcialg_2/D12')" onmouseout="rh()">BCIALG_2:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:69_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="bcialg_1.html#T2" onmouseover="rs('bcialg_1/T2')" onmouseout="rh()">BCIALG_1:2</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">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:69"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">y</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="E4:69"/>
( <font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="bcialg_1.html#T2" onmouseover="rs('bcialg_1/T2')" onmouseout="rh()">BCIALG_1:2</a>, <a class="ref" href="bcialg_1.html#D19" onmouseover="rs('bcialg_1/D19')" onmouseout="rh()">BCIALG_1:def 19</a></span>;<br/>
<a NAME="E5:69"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RK</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#D12" onmouseover="rs('bcialg_2/D12')" onmouseout="rh()">BCIALG_2:def 12</a></span>;<br/>
<a NAME="E6:69"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">RK</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" onmouseover="rs('eqrel_1/T18')" onmouseout="rh()">EQREL_1:18</a></span>;<br/>
<a NAME="E7:69"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">RK</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c3">RK</font>,<font color="Maroon" title="c7">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T23" onmouseover="rs('eqrel_1/T23')" onmouseout="rh()">EQREL_1:23</a></span>;<br/>
<a NAME="E8:69"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="bcialg_2.html#K9" title="BCIALG_2:func.9">./.</a> <font color="Maroon" title="c3">RK</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_6.html#D10" target="_self" onmouseover="rs('bcialg_6/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:69"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_6.html#K4" title="BCIALG_6:func.4">Ker</a> <span class="p1">(<span class="default"><a href="bcialg_6.html#K5" title="BCIALG_6:func.5">nat_hom</a> <font color="Maroon" title="c3">RK</font></span>)</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>
