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<span class="kw">set </span><font color="Maroon" title="c3">T</font> =  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>);<br/>
<span class="kw">thus </span><a NAME="E1:103_1_1"/>
 <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>) is <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</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">  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>) is <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</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="ref" href="proofs_1.html#D12" target="_self" onmouseover="rs('proofs_1/D12')" onmouseout="rh()">PROOFS_1:def 12</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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:103_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>)</span><br/>

<a NAME="E2:103_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#T8" target="_self" onmouseover="rs('proofs_1/T8')" onmouseout="rh()">Th46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:103_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#D34" target="_self" onmouseover="rs('proofs_1/D34')" onmouseout="rh()">Def30r</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="c4">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="proofs_1.html#D31" target="_self" onmouseover="rs('proofs_1/D31')" onmouseout="rh()">PROOFS_1:def 31</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">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Y</font>,<font color="Olive" title="b1">a</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="c2">R</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>) holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c5">a</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="c2">R</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>) implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:103_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c5">a</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="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E3:103_1_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</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="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>)</span><br/>

<a NAME="E4:103_1_1"/>
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">S</font> = <font color="Maroon" title="c4">Y</font> as   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#D1" target="_self" onmouseover="rs('proofs_1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E6:103_1_1"/>
 for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> holds <br/><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> implies <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:103_1_1_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</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="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E2:103_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c2">R</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:103_1_1_2"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> &amp; <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c8">t</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E5:103_1_1_2"/>
<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103_1_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></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="E7:103_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="proofs_1.html#T11" target="_self" onmouseover="rs('proofs_1/T11')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E8:103_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="xtuple_0.html#D13" onmouseover="rs('xtuple_0/D13')" onmouseout="rh()">XTUPLE_0:def 13</a></span>;<br/>
<a NAME="E9:103_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c2">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:103_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K11" title="PROOFS_1:func.11">Theorems</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:103_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></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>
