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<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/>

<a NAME="E1:67"/>
(  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_2.html#T45" target="_self" title="BCIALG_2:th.45">Th45</a>, <a class="ref" href="bcialg_2.html#T46" target="_self" title="BCIALG_2:th.46">Th46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:67"/>
 <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">thus </span><a NAME="E3:67"/>
<span class="p1">(<span class="default"><a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_2.html#K6" title="BCIALG_2:func.6">RConSet</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="bcialg_2.html#M3" title="BCIALG_2:mode.3">R-congruence</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">R</font>
 
;<br/>
<a NAME="E4:67_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_2.html#K5" title="BCIALG_2:func.5">LConSet</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:67_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">L</font> being   <a href="bcialg_2.html#M2" title="BCIALG_2:mode.2">L-congruence</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font>
 
;<br/>
<a NAME="E6:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> is    <a href="bcialg_2.html#M1" title="BCIALG_2:mode.1">Congruence</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="bcialg_2.html#T36" target="_self" title="BCIALG_2:th.36">Th36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:67_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bcialg_2.html#K4" title="BCIALG_2:func.4">ConSet</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


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