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<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"> ( <font color="Maroon" title="c1">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="classes1.html#V2" title="CLASSES1:attr.2">Tarski</a>  implies ( <font color="Maroon" title="c1">X</font> is  <a href="classes3.html#V2" title="CLASSES3:attr.2">union-closed</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="classes3.html#V3" title="CLASSES3:attr.3">FamUnion-closed</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_1_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="classes1.html#V2" title="CLASSES1:attr.2">Tarski</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> is  <a href="classes3.html#V2" title="CLASSES3:attr.2">union-closed</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="classes3.html#V3" title="CLASSES3:attr.3">FamUnion-closed</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:5_1_1"/>
<font color="Maroon" title="c1">X</font> is  <a href="classes3.html#V2" title="CLASSES3:attr.2">union-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="classes2.html#T59" onmouseover="rs('classes2/T59')" onmouseout="rh()">CLASSES2:59</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"> <font color="Maroon" title="c1">X</font> is  <a href="classes3.html#V3" title="CLASSES3:attr.3">FamUnion-closed</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">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="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="c2">Y</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> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</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> <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</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="c3">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<span class="kw">assume </span><a NAME="E3:5_1_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R2" title="TARSKI: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">  <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="c3">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c1">X</font> as   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:5_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E5:5_1_1"/>
(  not <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c4">X</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="classes2.html#T1" onmouseover="rs('classes2/T1')" onmouseout="rh()">CLASSES2:1</a></span>;<br/>
<a NAME="E6:5_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="ordinal1.html#T12" onmouseover="rs('ordinal1/T12')" onmouseout="rh()">ORDINAL1:12</a>, <a class="ref" href="card_1.html#T12" onmouseover="rs('card_1/T12')" onmouseout="rh()">CARD_1:12</a></span>;<br/>
<a NAME="E7:5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">X</font>
 
;<br/>
<a NAME="E8:5_1_1"/>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">X</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  or  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="classes1.html#D2" onmouseover="rs('classes1/D2')" onmouseout="rh()">CLASSES1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:5_1_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="c3">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_1_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a>, <a class="ref" href="classes2.html#T59" onmouseover="rs('classes2/T59')" onmouseout="rh()">CLASSES2:59</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/>


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