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

<span class="kw">set </span><font color="Maroon" title="c2">g</font> =  the    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$1 is    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">Gclasses</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:12_1_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Gclasses</font> iff ( <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xfamily.html#S1" onmouseover="rs('xfamily/S1')" onmouseout="rh()">XFAMILY:sch 1</a>();<br/></span>
<a NAME="E3:12_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 the    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D1" onmouseover="rs('zfmisc_1/D1')" onmouseout="rh()">ZFMISC_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Gclasses</font> = <font color="Maroon" title="c3">Gclasses</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">M</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font>;<br/>
<a NAME="E5:12_1_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_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#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> or <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font></span><br/>

<a NAME="E2:12_1_1_1"/>
 for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> holds <br/><font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c7">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> implies <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_1_1"/>
<font color="Maroon" title="c7">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font></span><br/>

<a NAME="E2:12_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c7">Z</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a>, <a class="txt" href="#E1:12_1_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_1_1_1_1"/>
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:12_1_1_1"/>
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T5" onmouseover="rs('setfam_1/T5')" onmouseout="rh()">SETFAM_1:5</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="E6:12_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> is  <a href="classes3.html#V1" title="CLASSES3:attr.1">power-closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">Z</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="classes3.html#D1" target="_self" onmouseover="rs('classes3/D1')" onmouseout="rh()">CLASSES3:def 1</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="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> implies  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_2"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</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="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font></span><br/>

<a NAME="E2:12_1_1_2"/>
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> holds <br/> <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Y</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"> ( <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> implies  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</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="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font></span><br/>

<a NAME="E2:12_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> is  <a href="classes3.html#V1" title="CLASSES3:attr.1">power-closed</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:12_1_1_2"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_1_1_2_1"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:12_1_1_2"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</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="E7:12_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> is  <a href="classes3.html#V3" title="CLASSES3:attr.3">FamUnion-closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">Z</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="classes3.html#D3" target="_self" onmouseover="rs('classes3/D3')" onmouseout="rh()">CLASSES3: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">  for <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> holds <br/> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:12_1_1_3"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> &amp; <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</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="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font></span><br/>

<a NAME="E2:12_1_1_3"/>
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> holds <br/> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">Y</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"> ( <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_3_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c8">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</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="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Y</font></span><br/>

<a NAME="E2:12_1_1_3_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
( <font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Y</font> &amp; <font color="Maroon" title="c8">Y</font> is  <a href="classes3.html#V3" title="CLASSES3:attr.3">FamUnion-closed</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:12_1_1_3"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E3:12_1_1_3_1"/>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> &amp;  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E1:12_1_1_3"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="setfam_1.html#T7" onmouseover="rs('setfam_1/T7')" onmouseout="rh()">SETFAM_1:7</a></span>;<br/>
<a NAME="E4:12_1_1_3_1"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:12_1_1_3_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_3"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E2:12_1_1_3_1"><span class="lab"><font color="Green" title="E12">A9</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">hence </span><a NAME="E3:12_1_1_3"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">M</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Gclasses</font> as   <a href="classes3.html#NM1" title="CLASSES3:NM.1">Grothendieck</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E6:12_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a></span>;<br/>
<a NAME="E9:12_1_1"/>
<font color="Maroon" title="c6">M</font> is    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1"><span class="kw">now </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">  for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> holds <br/><font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Y</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">Y</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"> ( <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</font> implies <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1_1_4_1"/>
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Gclasses</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font></span><br/><a NAME="E2:12_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">Y</font> is    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_1_1_4_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="classes3.html#D4" target="_self" onmouseover="rs('classes3/D4')" onmouseout="rh()">Def4</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">hence </span><a NAME="E2:12_1_1_4"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="classes3.html#D4" target="_self" onmouseover="rs('classes3/D4')" onmouseout="rh()">CLASSES3:def 4</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c6">M</font> as    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">M</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">G</font> being    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c7">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">G1</font> be    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</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"> <font color="Maroon" title="c7">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G1</font></span><br/>

<a NAME="E11:12_1_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<font color="Maroon" title="c7">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">G1</font> is    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes3.html#T1" target="_self" onmouseover="rs('classes3/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E12:12_1_1"/>
( <font color="Maroon" title="c7">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M</font> &amp; <font color="Maroon" title="c7">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T3" onmouseover="rs('setfam_1/T3')" onmouseout="rh()">SETFAM_1:3</a>, <a class="txt" href="#E3:12_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E13:12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A11</font></span>: 
( <font color="Maroon" title="c7">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the    <a href="classes3.html#M1" title="CLASSES3:mode.1">Grothendieck</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E14:12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="setfam_1.html#T3" onmouseover="rs('setfam_1/T3')" onmouseout="rh()">SETFAM_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:12_1_1"/>
<font color="Maroon" title="c7">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12_1_1"><span class="lab"><font color="Green" title="E11">A11</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>
