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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> holds <br/> ex <font color="Olive" title="b5">Y</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font> st <br/>( <font color="Olive" title="b5">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b5">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> )</span><br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> holds <br/> ex <font color="Olive" title="b4">Y</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b4">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/> ex <font color="Olive" title="b3">Y</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b3">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> implies  ex <font color="Olive" title="b1">Y</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> ) )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<a NAME="E3:19"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="closure1.html#T8" target="_self" title="CLOSURE1:th.8">Th8</a></i>;<br/>
<a NAME="E4:19"/>
<font color="Maroon" title="c3">X</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></i>;<br/>
<a NAME="E5:19"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T19" title="MBOOLEAN:th.19">MBOOLEAN:19</a></i>;<br/>
<a NAME="E6:19"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green" title="E16">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E8:19"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pzfmisc1.html#T1" title="PZFMISC1:th.1">PZFMISC1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">Y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:19"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E17">A6</font></i>: <font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E7:19"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<a NAME="E12:19"/>
<font color="Maroon" title="c7">Y</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D17" title="PBOOLE:def.17">PBOOLE:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )</span><br/>

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1"/>
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_1"><b>case </b></a><a NAME="E1:19_2_1_1_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span><br/><div class="add"><a NAME="E2:19_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="#E3:19"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E6:19"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_1"/>
<font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="#E11:19"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="#E1:19_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="#E1:19_2_1_1_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2"><b>case </b></a><a NAME="E1:19_2_1_1_2"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span><br/><div class="add"><a NAME="E2:19_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:19"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:19_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E4:19_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:19_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:19_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E6:19_2_1_1_2"/>
<font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="#E1:19_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_2"/>
<font color="Maroon" title="c7">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">Y</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font> ;<br/>
<a NAME="E14:19"/>
<font color="Maroon" title="c8">Y</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E1:19_3"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_3_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1_1"><b>case </b></a><a NAME="E1:19_3_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><div class="add"><b>hence </b><a NAME="E2:19_3_1_1_1"/>
<font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E11:19"><i><font color="Green" title="E17">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1_2"><b>case </b></a><a NAME="E1:19_3_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><div class="add"><a NAME="E2:19_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:19"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:19_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:19_3_1_1_2"/>
<font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E1:19_3"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_3"/>
<font color="Maroon" title="c8">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">Y</font> = <font color="Maroon" title="c8">Y</font> as  <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">Y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c9">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<b>thus </b><a NAME="E16:19"/>
<font color="Maroon" title="c9">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c9">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> )</span><br/>

<b>assume </b><a NAME="E1:19_4"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_1"><b>case </b></a><a NAME="E1:19_4_1_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><div class="add"><b>hence </b><a NAME="E2:19_4_1_1_1"/>
<font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="txt" href="#E3:19"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E11:19"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1_2"><b>case </b></a><a NAME="E1:19_4_1_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/><div class="add"><a NAME="E2:19_4_1_1_2"/><b>then </b>
not <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:19"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:19_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E4:19_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:19_4"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E5:19_4_1_1_2"/>
<font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_4"/>
<font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E17:19"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="#E11:19"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:19"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p3">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c9">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="closure1.html#T8" target="_self" title="CLOSURE1:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
