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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">R</font> being   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</a><br/>  for <font color="Olive" title="b2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b2">a</font> iff  for <font color="Olive" title="b3">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Olive">b<sub>1</sub></font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</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>  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">a</font> iff  for <font color="Olive" title="b2">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( <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="c3">a</font> iff  for <font color="Olive" title="b1">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:104"/>
( <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="c3">a</font> implies  for <font color="Olive" title="b1">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</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">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</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="c3">a</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:104_1"/><span class="lab"><font color="Green" title="E40">A1</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="c3">a</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">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">Y</font> be  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>;<br/>
<a NAME="E2:104_1"/>
<font color="Maroon" title="c4">Y</font> is <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a> 
 
;<br/>
<a NAME="E3:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 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="c1">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
;<br/>
<a NAME="E4:104_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">b</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="Olive" title="b1">Z</font>,<font color="Olive" title="b2">b</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; (  for <font color="Olive" title="b3">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">c</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">Z</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">b</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="c5">Z</font>,<font color="Olive" title="b1">b</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; (  for <font color="Olive" title="b2">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c6">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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c6">b</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; (  for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104_1_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">Z</font>,<font color="Maroon" title="c6">b</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="E2:104_1_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>]</span><br/>

<a NAME="E3:104_1_1"/>
<font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_1_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:104_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_1_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="proofs_1.html#D31" target="_self" onmouseover="rs('proofs_1/D31')" onmouseout="rh()">Def28</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="E5:104_1"/>
 for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="proofs_1.html#S3" target="_self" onmouseover="rs('proofs_1/S3')" onmouseout="rh()">PROOFS_1:sch 3</a>(<a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:104_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_1"><span class="lab"><font color="Green" title="E40">A1</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>

<span class="kw">assume </span><a NAME="E2:104"/>
 for <font color="Olive" title="b1">Y</font> being  <font color="Maroon" title="c1">X</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  <font color="Maroon" title="c2">R</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</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="c3">a</font></span><br/>

<a NAME="E3:104"/><span class="kw">then </span>
<font color="Maroon" title="c3">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>)
 
;<br/>
<span class="kw">hence </span><a NAME="E4:104"/>
<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="c3">a</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>
