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<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>  st  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">b</font> ) &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b2">a</font></span>)</span> ) holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  ex <font color="Olive" title="b3">phi</font> being  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b3">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b3">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b4">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b3">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">a</font> holds <br/>for <font color="Olive" title="b5">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b4">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b5">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b2">H</font> iff <font color="Olive" title="b1">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b2">H</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   <a href="zf_refle.html#NM2" title="ZF_REFLE:NM.2">DOMAIN-Sequence</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font> ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span> ) implies for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  ex <font color="Olive" title="b2">phi</font> being  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b3">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b4">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> ) ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font>
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  ex <font color="Olive" title="b2">phi</font> being  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b2">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b3">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">a</font> holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b4">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>;<br/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">phi</font> being  <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Olive" title="b1">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b3">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 ) ) );<br/>
<a NAME="E5:40"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#D5" target="_self" title="ZF_REFLE:def.5">Def5</a></span>;<br/>
<a NAME="E6:40"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>,  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a>) <span class="kw">-&gt; </span>  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> =  <a href="zf_refle.html#K2" title="ZF_REFLE:func.2">union</a> $2,$1;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">x0</font> =  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">H9</font> =  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) );<br/>
<span class="kw">assume </span><a NAME="E1:40_1"/>
<font color="Maroon" title="c4">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<a NAME="E2:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a></span>;<br/>
<a NAME="E3:40_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
for <font color="Olive" title="b1">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span>  ex <font color="Olive" title="b2">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">h</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Olive" title="b1">a</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">h</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1_1_1"/>
( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> or  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1_1_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Olive" title="b1">a</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Olive" title="b1">a</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:40_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Olive" title="b1">a</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E2:40_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_1_1_2"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <span class="kw">and </span><br/><a NAME="E5:40_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:40_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c12">x</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> <font color="Maroon" title="c2">L</font>
 <span class="kw">and </span><br/><a NAME="E8:40_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c13">x</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:40_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Maroon" title="c14">b</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> &amp; (  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E11:40_1_1_1_1_2_1"/>
( <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> &amp; (  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c9">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E4:40_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E8:40_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:40_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">h</font>,<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">rho</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:40_1"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">rho</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:40_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
for <font color="Olive" title="b1">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span>  ex <font color="Olive" title="b2">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">a</font>] &amp; ( for <font color="Olive" title="b3">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">h</font>,<font color="Olive" title="b3">b</font>] holds <br/><font color="Olive" title="b2">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b3">b</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S1" target="_self" title="ZF_REFLE:sch.1">ZF_REFLE:sch 1</a>(<a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>,  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> $1</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E7:40_1"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  ex <font color="Olive" title="b2">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">b</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:40_1_2"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:40_1_2_1"/>
<font color="Maroon" title="c11">h</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> <font color="Maroon" title="c7">rho</font>
 <span class="kw">and </span><br/><a NAME="E4:40_1_2_1"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c11">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:40_1_2_1"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E6:40_1_2_1"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="funct_5.html#T63" title="FUNCT_5:th.63">FUNCT_5:63</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c11">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_2_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a></span>;<br/>
<a NAME="E8:40_1_2_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">h</font> &amp;  ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> &amp; (  ex <font color="Olive" title="b4">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b2">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b3">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b4">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b4">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b3">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b4">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Olive" title="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">b</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:40_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_2_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:40_1_2"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="classes2.html#T67" title="CLASSES2:th.67">CLASSES2:67</a></span>;<br/>
<a NAME="E3:40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></span>;<br/>
<a NAME="E4:40_1_2"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T25" target="_self" title="ZF_REFLE:th.25">Th25</a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></span>;<br/>
<a NAME="E5:40_1_2"/><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="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T68" title="FUNCT_5:th.68">FUNCT_5:68</a></span>;<br/>
<a NAME="E6:40_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="card_2.html#T3" title="CARD_2:th.3">CARD_2:3</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E7:40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">b</font> =  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T28" target="_self" title="ZF_REFLE:th.28">Th28</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">b</font>]</span><br/>

<span class="kw">thus </span><a NAME="E9:40_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">b</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:40_1"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b2</font>] holds <br/><font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b2</font>
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">si</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E10:40_1"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S2" target="_self" title="ZF_REFLE:sch.2">ZF_REFLE:sch 2</a>(<a class="txt" href="#E8:40_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="txt" href="#E7:40_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>,  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> =  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> $1</span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> $2</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">ksi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E12:40_1"/><span class="lab"><font color="Green" title="E30">A24</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:40_1"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font>)
 <span class="kw">and </span><br/><a NAME="E14:40_1"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S3" target="_self" title="ZF_REFLE:sch.3">ZF_REFLE:sch 3</a>();<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span><font color="Maroon" title="c8">si</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">given </span><font color="Maroon" title="c10">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><a NAME="E16:40_1"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c10">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E17:40_1"/><span class="lab"><font color="Green" title="E34">A28</font></span>: 
<font color="Maroon" title="c10">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E18:40_1"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]</span><br/>

<a NAME="E19:40_1"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
<font color="Maroon" title="c9">ksi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">ksi</font> or <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> or  not <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">ksi</font> or <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">B</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_3"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_3"/><span class="lab"><font color="Green" title="E38">A32</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">B</font></span><br/>

<a NAME="E3:40_1_3"/>
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="txt" href="#E2:40_1_3"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">A</font>, <font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c12">B</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>] <span class="kw">means </span>( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 implies <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> $1 );<br/>
<a NAME="E5:40_1_3"/><span class="lab"><font color="Green" title="E39">A33</font></span>: 
for <font color="Olive" title="b1">c</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">c</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">c</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c15">c</font>] implies <font color="Maroon">S<sub>5</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_3_1"/><span class="lab"><font color="Green" title="E40">A34</font></span>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">c</font> implies <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 <span class="kw">and </span><br/><a NAME="E2:40_1_3_1"/><span class="lab"><font color="Green" title="E41">A35</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span><br/>

<a NAME="E3:40_1_3_1"/><span class="lab"><font color="Green" title="E42">A36</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_3_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/>
<a NAME="E4:40_1_3_1"/><span class="lab"><font color="Green" title="E43">A37</font></span>: 
( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_3_1_1_1"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_3_1_1_1_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )</span><br/><div class="add"><a NAME="E2:40_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">a</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c15">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_3_1"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_3_1_1_1_1"/>
( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3_1"><span class="lab"><font color="Green" title="E40">A34</font></span></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_3_1_1_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_3_1_1_1_2"/>
( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> or <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:40_1_3_1"/><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:40_1_3_1"/>
( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40_1_3_1"/>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_3_1"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="txt" href="#E4:40_1_3_1"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:40_1_3"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">c</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">c</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:40_1_3_2"/>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a></span>;<br/>
<a NAME="E2:40_1_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E3:40_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A40</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">b</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c15">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">c</font>] ) implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c15">b</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:40_1_3_2"/><span class="lab"><font color="Green" title="E42">A41</font></span>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E5:40_1_3_2"/><span class="lab"><font color="Green" title="E43">A42</font></span>: 
<font color="Maroon" title="c15">b</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">and </span><br/><a NAME="E6:40_1_3_2"/>
for <font color="Olive" title="b1">c</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">c</font>]
 <span class="kw">and </span><br/><a NAME="E7:40_1_3_2"/><span class="lab"><font color="Green" title="E44">A43</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">b</font></span><br/>

<a NAME="E8:40_1_3_2"/>
(  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_3_2"><span class="lab"><font color="Green" title="E43">A42</font></span></a>, <a class="txt" href="#E7:40_1_3_2"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<a NAME="E9:40_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A44</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></span>;<br/>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">b</font> = 
 <a href="zf_refle.html#K2" title="ZF_REFLE:func.2">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span>,<font color="Maroon" title="c15">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E4:40_1_3_2"><span class="lab"><font color="Green" title="E42">A41</font></span></a>, <a class="txt" href="#E5:40_1_3_2"><span class="lab"><font color="Green" title="E43">A42</font></span></a></span>
<br/>.= 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T17" target="_self" title="ZF_REFLE:th.17">Th17</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:40_1_3_2"/>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_3_2"><span class="lab"><font color="Green" title="E45">A44</font></span></a>, <a class="txt" href="#E3:40_1_3_2"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:40_1_3"/><span class="lab"><font color="Green" title="E41">A45</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a></span>;<br/>
<a NAME="E8:40_1_3"/>
for <font color="Olive" title="b1">c</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">c</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S4" target="_self" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="#E7:40_1_3"><span class="lab"><font color="Green" title="E41">A45</font></span></a>, <a class="txt" href="#E5:40_1_3"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="txt" href="#E6:40_1_3"><span class="lab"><font color="Green" title="E40">A39</font></span></a>);<br/></span>
<a NAME="E9:40_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c14">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E37">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:40_1_3"/>
<font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:40_1"/><span class="lab"><font color="Green" title="E37">A46</font></span>: 
 <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T35" title="ORDINAL4:th.35">ORDINAL4:35</a></span>;<br/>
<a NAME="E21:40_1"/><span class="lab"><font color="Green" title="E38">A47</font></span>: 
<font color="Maroon" title="c9">ksi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">ksi</font> or <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c11">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> or <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">ksi</font> or <font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c11">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font> or <font color="Maroon" title="c12">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_4"/><span class="lab"><font color="Green" title="E39">A48</font></span>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_4"/><span class="lab"><font color="Green" title="E40">A49</font></span>: 
<font color="Maroon" title="c11">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:40_1_4"/><span class="lab"><font color="Green" title="E41">A50</font></span>: 
<font color="Maroon" title="c11">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">and </span><br/><a NAME="E4:40_1_4"/><span class="lab"><font color="Green" title="E42">A51</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ksi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span><br/>

<a NAME="E5:40_1_4"/>
<font color="Maroon" title="c11">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E39">A48</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E6:40_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A52</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c11">A</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E39">A48</font></span></a>, <a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/>
<span class="lab"><font color="Green" title="E44">A53</font></span>: <font color="Maroon" title="c12">B</font> = 
 <a href="zf_refle.html#K2" title="ZF_REFLE:func.2">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">a</font></span>)</span>,<font color="Maroon" title="c13">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E20:40_1"><span class="lab"><font color="Green" title="E37">A46</font></span></a>, <a class="txt" href="#E2:40_1_4"><span class="lab"><font color="Green" title="E40">A49</font></span></a>, <a class="txt" href="#E3:40_1_4"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E4:40_1_4"><span class="lab"><font color="Green" title="E42">A51</font></span></a></span>
<br/>.= 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T17" target="_self" title="ZF_REFLE:th.17">Th17</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>
;<br/>
<a NAME="E9:40_1_4"/><span class="lab"><font color="Green" title="E45">A54</font></span>: 
<font color="Maroon" title="c12">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> or <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_4_2"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:40_1_4_2"/><span class="lab"><font color="Green" title="E46">A55</font></span>: 
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font>
 <span class="kw">and </span><br/><a NAME="E4:40_1_4_2"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
<font color="Maroon" title="c15">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_4"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:40_1_4_2"/>
 ex <font color="Olive" title="b1">A1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">X</font> = <font color="Maroon" title="c15">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_4_2"><span class="lab"><font color="Green" title="E47">A56</font></span></a></span>;<br/>
<a NAME="E7:40_1_4_2"/>
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_4_2"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40_1_4_2"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4_2"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:40_1_4"/><span class="lab"><font color="Green" title="E46">A57</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/>
<a NAME="E11:40_1_4"/><span class="lab"><font color="Green" title="E47">A58</font></span>: 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c12">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> or <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_4_3"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:40_1_4_3"/><span class="lab"><font color="Green" title="E48">A59</font></span>: 
<font color="Maroon" title="c15">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:40_1_4_3"/><span class="lab"><font color="Green" title="E49">A60</font></span>: 
<font color="Maroon" title="c14">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c15">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:40_1_4_3"/><span class="lab"><font color="Green" title="E50">A61</font></span>: 
<font color="Maroon" title="c16">x</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="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:40_1_4_3"/><span class="lab"><font color="Green" title="E51">A62</font></span>: 
<font color="Maroon" title="c15">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4_3"><span class="lab"><font color="Green" title="E48">A59</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_4_3"><span class="lab"><font color="Green" title="E50">A61</font></span></a></span>;<br/>
<a NAME="E9:40_1_4_3"/><span class="lab"><font color="Green" title="E52">A63</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_4"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E6:40_1_4"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E6:40_1_4_3"><span class="lab"><font color="Green" title="E50">A61</font></span></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></span>;<br/>
<a NAME="E10:40_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A64</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_4"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E11:40_1_4_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E12:40_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c17">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_4"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E10:40_1_4"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="txt" href="#E7:40_1_4_3"><span class="lab"><font color="Green" title="E51">A62</font></span></a>, <a class="txt" href="#E9:40_1_4_3"><span class="lab"><font color="Green" title="E52">A63</font></span></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></span>;<br/>
<a NAME="E13:40_1_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_4"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="txt" href="#E10:40_1_4_3"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:40_1_4_3"/>
<font color="Maroon" title="c14">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_4_3"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:40_1_4"/>
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font></span>)</span> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_4"><span class="lab"><font color="Green" title="E40">A49</font></span></a>, <a class="txt" href="#E3:40_1_4"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E6:40_1_4"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E10:40_1_4"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:40_1_4"/>
<font color="Maroon" title="c12">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_4"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="txt" href="#E11:40_1_4"><span class="lab"><font color="Green" title="E47">A58</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:40_1"/><span class="lab"><font color="Green" title="E39">A65</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  holds <br/><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c11">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  implies <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_5"/><span class="lab"><font color="Green" title="E40">A66</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_5"/><span class="lab"><font color="Green" title="E41">A67</font></span>: 
<font color="Maroon" title="c11">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span><br/>

<a NAME="E3:40_1_5"/><span class="lab"><font color="Green" title="E42">A68</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> or <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:40_1_5"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:40_1_5"/><span class="lab"><font color="Green" title="E43">A69</font></span>: 
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:40_1_5"/><span class="lab"><font color="Green" title="E44">A70</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_5"><span class="lab"><font color="Green" title="E42">A68</font></span></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:40_1_5"/><span class="lab"><font color="Green" title="E45">A71</font></span>: 
<font color="Maroon" title="c14">x</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> <font color="Maroon" title="c7">rho</font>
 <span class="kw">and </span><br/><a NAME="E10:40_1_5"/><span class="lab"><font color="Green" title="E46">A72</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:40_1_5"/><span class="lab"><font color="Green" title="E47">A73</font></span>: 
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_5"><span class="lab"><font color="Green" title="E43">A69</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E9:40_1_5"><span class="lab"><font color="Green" title="E45">A71</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">a1</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E14:40_1_5"/><span class="lab"><font color="Green" title="E48">A74</font></span>: 
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">h</font>
 <span class="kw">and </span><br/><a NAME="E15:40_1_5"/><span class="lab"><font color="Green" title="E49">A75</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <span class="kw">and </span><br/><a NAME="E16:40_1_5"/><span class="lab"><font color="Green" title="E50">A76</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Maroon" title="c16">a1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:40_1_5"/><span class="lab"><font color="Green" title="E51">A77</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font>
 <span class="kw">and </span><br/><a NAME="E19:40_1_5"/><span class="lab"><font color="Green" title="E52">A78</font></span>: 
(  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40_1_5"><span class="lab"><font color="Green" title="E49">A75</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>6</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>;<br/>
<div><span class="lab"><font color="Green" title="E53">A79</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:40_1_5_1"/><span class="lab"><font color="Green" title="E54">A80</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>( <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">y</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c18">x</font>] )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c20">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><a NAME="E2:40_1_5_1_1"/><span class="lab"><font color="Green" title="E55">A81</font></span>: 
( <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">a</font> &amp; not <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c18">x</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c21">b</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font> implies <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:40_1_5_1_1"/><span class="lab"><font color="Green" title="E56">A82</font></span>: 
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font></span><br/>

<a NAME="E4:40_1_5_1_1"/>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b</font> or <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b</font> or <font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></span>;<br/>
<a NAME="E5:40_1_5_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font> or <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c20">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_1_5_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c21">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_5_1_1"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="txt" href="#E3:40_1_5_1_1"><span class="lab"><font color="Green" title="E56">A82</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:40_1_5_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><a NAME="E3:40_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A83</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:40_1_5_1"/><span class="lab"><font color="Green" title="E56">A84</font></span>: 
( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E20:40_1"><span class="lab"><font color="Green" title="E37">A46</font></span></a>, <a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E40">A66</font></span></a>, <a class="txt" href="#E2:40_1_5"><span class="lab"><font color="Green" title="E41">A67</font></span></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/><a NAME="E5:40_1_5_1"/><span class="lab"><font color="Green" title="E57">A85</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:40_1_5_1"/><span class="lab"><font color="Green" title="E58">A86</font></span>: 
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c18">z</font>,<font color="Olive" title="b1">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="card_3.html#S3" title="CARD_3:sch.3">CARD_3:sch 3</a>(<a class="txt" href="#E3:40_1_5_1"><span class="lab"><font color="Green" title="E55">A83</font></span></a>, <a class="txt" href="#E1:40_1_5_1"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="txt" href="#E5:40_1_5_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c19">z</font> = <font color="Maroon" title="c18">z</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_5_1"><span class="lab"><font color="Green" title="E58">A86</font></span></a></span>;<br/><a NAME="E9:40_1_5_1"/><span class="lab"><font color="Green" title="E59">A87</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></span>;<br/><a NAME="E10:40_1_5_1"/><span class="lab"><font color="Green" title="E60">A88</font></span>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1_5"><span class="lab"><font color="Green" title="E46">A72</font></span></a>, <a class="txt" href="#E18:40_1_5"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><a NAME="E11:40_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:40_1_5_1"/><span class="lab"><font color="Green" title="E61">A89</font></span>: 
<font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X</font>
 <span class="kw">and </span><br/><a NAME="E14:40_1_5_1"/><span class="lab"><font color="Green" title="E62">A90</font></span>: 
<font color="Maroon" title="c20">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1_5_1"><span class="lab"><font color="Green" title="E60">A88</font></span></a>, <a class="txt" href="#E4:40_1_5_1"><span class="lab"><font color="Green" title="E56">A84</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:40_1_5_1"/><span class="lab"><font color="Green" title="E63">A91</font></span>: 
<font color="Maroon" title="c21">x</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="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:40_1_5_1"/><span class="lab"><font color="Green" title="E64">A92</font></span>: 
<font color="Maroon" title="c20">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_5_1"><span class="lab"><font color="Green" title="E62">A90</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E18:40_1_5_1"/><span class="lab"><font color="Green" title="E65">A93</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1_5_1"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c21">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1_5_1"><span class="lab"><font color="Green" title="E63">A91</font></span></a></span>;<br/><a NAME="E20:40_1_5_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/><a NAME="E21:40_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1_5_1"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="txt" href="#E9:40_1_5_1"><span class="lab"><font color="Green" title="E59">A87</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c22">x</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c23">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c24">x</font> ) )</span><br/><span class="kw">thus </span><a NAME="E23:40_1_5_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1_5_1"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="txt" href="#E9:40_1_5_1"><span class="lab"><font color="Green" title="E59">A87</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c24">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c25">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> implies <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c24">x</font> )</span><br/><span class="kw">assume </span><a NAME="E24:40_1_5_1"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c24">x</font></span><br/><span class="kw">hence </span><a NAME="E25:40_1_5_1"/>
<font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_5_1"><span class="lab"><font color="Green" title="E58">A86</font></span></a>, <a class="txt" href="#E13:40_1_5_1"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="txt" href="#E17:40_1_5_1"><span class="lab"><font color="Green" title="E64">A92</font></span></a>, <a class="txt" href="#E18:40_1_5_1"><span class="lab"><font color="Green" title="E65">A93</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_2"/><span class="lab"><font color="Green" title="E54">A94</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; ( for <font color="Olive" title="b2">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c18">b</font> =  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> &amp; ( for <font color="Olive" title="b1">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">b</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:40_1_5_2"/>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40_1"><span class="lab"><font color="Green" title="E37">A46</font></span></a>, <a class="txt" href="#E1:40_1_5"><span class="lab"><font color="Green" title="E40">A66</font></span></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">b</font></span><br/><span class="kw">thus </span><a NAME="E3:40_1_5_2"/>
for <font color="Olive" title="b1">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5_2"><span class="lab"><font color="Green" title="E54">A94</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">consider </span><font color="Maroon" title="c18">c</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E23:40_1_5"/><span class="lab"><font color="Green" title="E54">A95</font></span>: 
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">and </span><br/><a NAME="E24:40_1_5"/><span class="lab"><font color="Green" title="E55">A96</font></span>: 
for <font color="Olive" title="b1">x1</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:40_1_5"><span class="lab"><font color="Green" title="E53">A79</font></span></a></span>;<br/>
<a NAME="E25:40_1_5"/><span class="lab"><font color="Green" title="E56">A97</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a></span>;<br/>
<a NAME="E26:40_1_5"/>
( <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">si</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">si</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c11">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E27:40_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A98</font></span>: 
<font color="Maroon" title="c18">c</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="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40_1_5"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E28:40_1_5"/>
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40_1_5"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E29:40_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40_1_5"><span class="lab"><font color="Green" title="E57">A98</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E30:40_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A99</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c19">e</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">v</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E33:40_1_5"/><span class="lab"><font color="Green" title="E59">A100</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  holds <br/>( ( <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c20">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; (  not <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c20">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">x</font>) ) ) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></span>
<a NAME="E34:40_1_5"/><span class="lab"><font color="Green" title="E60">A101</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">v</font> or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_5_3"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:40_1_5_3"/><span class="lab"><font color="Green" title="E61">A102</font></span>: 
<font color="Maroon" title="c22">y</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> <font color="Maroon" title="c20">v</font>
 <span class="kw">and </span><br/><a NAME="E4:40_1_5_3"/><span class="lab"><font color="Green" title="E62">A103</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c22">y</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E3:40_1_5_3"><span class="lab"><font color="Green" title="E61">A102</font></span></a></span>;<br/>
<a NAME="E6:40_1_5_3"/>
( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> or  not <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E7:40_1_5_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">y</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font> ) or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:40_1_5"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E4:40_1_5_3"><span class="lab"><font color="Green" title="E62">A103</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40_1_5_3"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c21">v</font> = <font color="Maroon" title="c20">v</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<a NAME="E36:40_1_5"/><span class="lab"><font color="Green" title="E61">A104</font></span>: 
<font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E34:40_1_5"><span class="lab"><font color="Green" title="E60">A101</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E37:40_1_5"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="funct_5.html#T63" title="FUNCT_5:th.63">FUNCT_5:63</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">v9</font> = <font color="Maroon" title="c21">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:40_1_5"><span class="lab"><font color="Green" title="E61">A104</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">a2</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E40:40_1_5"/><span class="lab"><font color="Green" title="E62">A105</font></span>: 
<font color="Maroon" title="c23">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v9</font>
 <span class="kw">and </span><br/><a NAME="E41:40_1_5"/><span class="lab"><font color="Green" title="E63">A106</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c22">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c23">a2</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c23">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <span class="kw">and </span><br/><a NAME="E42:40_1_5"/><span class="lab"><font color="Green" title="E64">A107</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c22">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Maroon" title="c23">a2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">f9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E44:40_1_5"/><span class="lab"><font color="Green" title="E65">A108</font></span>: 
<font color="Maroon" title="c22">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f9</font>
 <span class="kw">and </span><br/><a NAME="E45:40_1_5"/><span class="lab"><font color="Green" title="E66">A109</font></span>: 
(  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c23">a2</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c24">f9</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c23">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c24">f9</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:40_1_5"><span class="lab"><font color="Green" title="E63">A106</font></span></a></span>;<br/>
<a NAME="E46:40_1_5"/><span class="lab"><font color="Green" title="E67">A110</font></span>: 
<font color="Maroon" title="c21">v</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E68">A111</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_4"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c25">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><a NAME="E2:40_1_5_4"/><span class="lab"><font color="Green" title="E69">A112</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font>
 <span class="kw">and </span><br/><a NAME="E3:40_1_5_4"/><span class="lab"><font color="Green" title="E70">A113</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2</font></span><br/><div><span class="lab"><font color="Green" title="E71">A114</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">x1</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> )</span><br/><a NAME="E1:40_1_5_4_1"/><span class="lab"><font color="Green" title="E72">A115</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>;<br/><a NAME="E2:40_1_5_4_1"/><span class="lab"><font color="Green" title="E73">A116</font></span>: 
( <font color="Maroon" title="c26">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>;<br/><span class="kw">assume </span><a NAME="E3:40_1_5_4_1"/>
<font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font></span><br/><span class="kw">hence </span><a NAME="E4:40_1_5_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E46:40_1_5"><span class="lab"><font color="Green" title="E67">A110</font></span></a>, <a class="txt" href="#E1:40_1_5_4_1"><span class="lab"><font color="Green" title="E72">A115</font></span></a>, <a class="txt" href="#E2:40_1_5_4_1"><span class="lab"><font color="Green" title="E73">A116</font></span></a>, <a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:40_1_5_4"/><span class="kw">then </span>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_5_4"><span class="lab"><font color="Green" title="E70">A113</font></span></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">m9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E7:40_1_5_4"/><span class="lab"><font color="Green" title="E72">A117</font></span>: 
( <font color="Maroon" title="c26">m9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c23">a2</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c24">f9</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40_1_5"><span class="lab"><font color="Green" title="E67">A110</font></span></a>, <a class="txt" href="#E44:40_1_5"><span class="lab"><font color="Green" title="E65">A108</font></span></a>, <a class="txt" href="#E45:40_1_5"><span class="lab"><font color="Green" title="E66">A109</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">x1</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f9</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> )</span><br/><a NAME="E1:40_1_5_4_2"/><span class="lab"><font color="Green" title="E73">A118</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>;<br/><a NAME="E2:40_1_5_4_2"/><span class="lab"><font color="Green" title="E74">A119</font></span>: 
( <font color="Maroon" title="c27">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>;<br/><span class="kw">assume </span><a NAME="E3:40_1_5_4_2"/>
<font color="Maroon" title="c27">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f9</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font></span><br/><span class="kw">hence </span><a NAME="E4:40_1_5_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f9</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c26">m9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E46:40_1_5"><span class="lab"><font color="Green" title="E67">A110</font></span></a>, <a class="txt" href="#E44:40_1_5"><span class="lab"><font color="Green" title="E65">A108</font></span></a>, <a class="txt" href="#E1:40_1_5_4_2"><span class="lab"><font color="Green" title="E73">A118</font></span></a>, <a class="txt" href="#E2:40_1_5_4_2"><span class="lab"><font color="Green" title="E74">A119</font></span></a>, <a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:40_1_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A120</font></span>: 
<font color="Maroon" title="c16">a1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c23">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1_5"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E18:40_1_5"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="txt" href="#E7:40_1_5_4"><span class="lab"><font color="Green" title="E72">A117</font></span></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></span>;<br/><a NAME="E10:40_1_5_4"/>
<font color="Maroon" title="c23">a2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c16">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:40_1_5"><span class="lab"><font color="Green" title="E67">A110</font></span></a>, <a class="txt" href="#E42:40_1_5"><span class="lab"><font color="Green" title="E64">A107</font></span></a>, <a class="txt" href="#E2:40_1_5_4"><span class="lab"><font color="Green" title="E69">A112</font></span></a>, <a class="txt" href="#E3:40_1_5_4"><span class="lab"><font color="Green" title="E70">A113</font></span></a>, <a class="txt" href="#E4:40_1_5_4"><span class="lab"><font color="Green" title="E71">A114</font></span></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></span>;<br/><span class="kw">hence </span><a NAME="E11:40_1_5_4"/>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_5_4"><span class="lab"><font color="Green" title="E73">A120</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_5_5"/><span class="lab"><font color="Green" title="E69">A121</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b1">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_5_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">x1</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> )</span><br/><a NAME="E1:40_1_5_5_1_1"/><span class="lab"><font color="Green" title="E70">A122</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>;<br/><a NAME="E2:40_1_5_5_1_1"/><span class="lab"><font color="Green" title="E71">A123</font></span>: 
( <font color="Maroon" title="c26">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>;<br/><span class="kw">assume </span><a NAME="E3:40_1_5_5_1_1"/>
<font color="Maroon" title="c26">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font></span><br/><span class="kw">hence </span><a NAME="E4:40_1_5_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:40_1_5"><span class="lab"><font color="Green" title="E59">A100</font></span></a>, <a class="txt" href="#E46:40_1_5"><span class="lab"><font color="Green" title="E67">A110</font></span></a>, <a class="txt" href="#E1:40_1_5_5_1_1"><span class="lab"><font color="Green" title="E70">A122</font></span></a>, <a class="txt" href="#E2:40_1_5_5_1_1"><span class="lab"><font color="Green" title="E71">A123</font></span></a>, <a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:40_1_5_5_1"/>
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c21">v</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c25">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_5_5"><span class="lab"><font color="Green" title="E69">A121</font></span></a>, <a class="ref" href="zf_lang1.html#T84" title="ZF_LANG1:th.84">ZF_LANG1:84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:40_1_5_5"/>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1_5"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="txt" href="#E46:40_1_5"><span class="lab"><font color="Green" title="E67">A110</font></span></a>, <a class="txt" href="#E44:40_1_5"><span class="lab"><font color="Green" title="E65">A108</font></span></a>, <a class="txt" href="#E45:40_1_5"><span class="lab"><font color="Green" title="E66">A109</font></span></a>, <a class="txt" href="#E1:40_1_5_5"><span class="lab"><font color="Green" title="E69">A121</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E49:40_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E11:40_1_5"><span class="lab"><font color="Green" title="E47">A73</font></span></a>, <a class="txt" href="#E14:40_1_5"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E15:40_1_5"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E18:40_1_5"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="txt" href="#E36:40_1_5"><span class="lab"><font color="Green" title="E61">A104</font></span></a>, <a class="txt" href="#E40:40_1_5"><span class="lab"><font color="Green" title="E62">A105</font></span></a>, <a class="txt" href="#E47:40_1_5"><span class="lab"><font color="Green" title="E68">A111</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E50:40_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c18">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40_1_5"><span class="lab"><font color="Green" title="E56">A97</font></span></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<a NAME="E51:40_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:40_1_5"><span class="lab"><font color="Green" title="E58">A99</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E52:40_1_5"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_5"><span class="lab"><font color="Green" title="E44">A70</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:40_1"/><span class="lab"><font color="Green" title="E40">A124</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Olive" title="b1">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c11">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">b</font>] ) implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">a</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_6"/><span class="lab"><font color="Green" title="E41">A125</font></span>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c11">a</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E2:40_1_6"/><span class="lab"><font color="Green" title="E42">A126</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">a</font> holds <br/><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">a</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:40_1_6"/>
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:40_1_6_1"/><span class="lab"><font color="Green" title="E43">A127</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:40_1"><span class="lab"><font color="Green" title="E39">A65</font></span></a>, <a class="txt" href="#E1:40_1_6"><span class="lab"><font color="Green" title="E41">A125</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal1.html#D5" title="ORDINAL1:def.5">ORDINAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> or <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:40_1_6_1"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E4:40_1_6_1"/><span class="lab"><font color="Green" title="E44">A128</font></span>: 
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:40_1_6_1"/><span class="lab"><font color="Green" title="E45">A129</font></span>: 
<font color="Maroon" title="c12">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_6_1"><span class="lab"><font color="Green" title="E43">A127</font></span></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:40_1_6_1"/><span class="lab"><font color="Green" title="E46">A130</font></span>: 
<font color="Maroon" title="c14">x</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="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:40_1_6_1"/><span class="lab"><font color="Green" title="E47">A131</font></span>: 
<font color="Maroon" title="c13">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_6_1"><span class="lab"><font color="Green" title="E44">A128</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E9:40_1_6_1"/><span class="lab"><font color="Green" title="E48">A132</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_6_1"><span class="lab"><font color="Green" title="E46">A130</font></span></a></span>;<br/>
<a NAME="E11:40_1_6_1"/><span class="lab"><font color="Green" title="E49">A133</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/>
<a NAME="E12:40_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_6_1"><span class="lab"><font color="Green" title="E46">A130</font></span></a>, <a class="txt" href="#E9:40_1_6_1"><span class="lab"><font color="Green" title="E48">A132</font></span></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font> as   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D10" title="ORDINAL1:def.10">ORDINAL1:def 10</a></span>;<br/>
<a NAME="E14:40_1_6_1"/><span class="lab"><font color="Green" title="E50">A134</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_6_1"><span class="lab"><font color="Green" title="E46">A130</font></span></a>, <a class="txt" href="#E8:40_1_6_1"><span class="lab"><font color="Green" title="E47">A131</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E15:40_1_6_1"/><span class="lab"><font color="Green" title="E51">A135</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_6"><span class="lab"><font color="Green" title="E42">A126</font></span></a>, <a class="txt" href="#E7:40_1_6_1"><span class="lab"><font color="Green" title="E46">A130</font></span></a>, <a class="txt" href="#E9:40_1_6_1"><span class="lab"><font color="Green" title="E48">A132</font></span></a></span>;<br/>
<a NAME="E16:40_1_6_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E17:40_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E7:40_1_6_1"><span class="lab"><font color="Green" title="E46">A130</font></span></a>, <a class="txt" href="#E9:40_1_6_1"><span class="lab"><font color="Green" title="E48">A132</font></span></a>, <a class="txt" href="#E11:40_1_6_1"><span class="lab"><font color="Green" title="E49">A133</font></span></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:40_1_6_1"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_6_1"><span class="lab"><font color="Green" title="E45">A129</font></span></a>, <a class="txt" href="#E14:40_1_6_1"><span class="lab"><font color="Green" title="E50">A134</font></span></a>, <a class="txt" href="#E15:40_1_6_1"><span class="lab"><font color="Green" title="E51">A135</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E24:40_1"/><span class="lab"><font color="Green" title="E41">A136</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">a</font>] implies <font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_7"/>
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font>]</span><br/>

<a NAME="E2:40_1_7"/>
( <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p2">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p3">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="zf_refle.html#K3" title="ZF_REFLE:func.3">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/>
<a NAME="E3:40_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_1_7"/>
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:40_1"/><span class="lab"><font color="Green" title="E42">A137</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a></span>;<br/>
<a NAME="E26:40_1"/><span class="lab"><font color="Green" title="E43">A138</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="zf_refle.html#S4" target="_self" title="ZF_REFLE:sch.4">ZF_REFLE:sch 4</a>(<a class="txt" href="#E25:40_1"><span class="lab"><font color="Green" title="E42">A137</font></span></a>, <a class="txt" href="#E24:40_1"><span class="lab"><font color="Green" title="E41">A136</font></span></a>, <a class="txt" href="#E23:40_1"><span class="lab"><font color="Green" title="E40">A124</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E44">A139</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_8"/>
 <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]</span><br/><span class="kw">thus </span><a NAME="E2:40_1_8"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c11">gamma</font> = <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c9">ksi</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">gamma</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c11">gamma</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c11">gamma</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) ) )</span><br/>

<a NAME="E1:40_1_8_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c9">ksi</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E19:40_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="ordinal4.html#T13" title="ORDINAL4:th.13">ORDINAL4:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:40_1_8_1"/>
( <font color="Maroon" title="c11">gamma</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c11">gamma</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40_1"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E19:40_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E21:40_1"><span class="lab"><font color="Green" title="E38">A47</font></span></a>, <a class="ref" href="ordinal4.html#T17" title="ORDINAL4:th.17">ORDINAL4:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c11">gamma</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">gamma</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">a</font> implies for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:40_1_8_1"/><span class="lab"><font color="Green" title="E45">A140</font></span>: 
<font color="Maroon" title="c11">gamma</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">and </span><br/><a NAME="E4:40_1_8_1"/><span class="lab"><font color="Green" title="E46">A141</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<a NAME="E5:40_1_8_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">gamma</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<a NAME="E6:40_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A142</font></span>: 
<font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">gamma</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E7:40_1_8_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">ksi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<a NAME="E8:40_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A143</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<a NAME="E9:40_1_8_1"/>
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<a NAME="E10:40_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A144</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<a NAME="E11:40_1_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A145</font></span>: 
<font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_8_1"><span class="lab"><font color="Green" title="E45">A140</font></span></a>, <a class="txt" href="#E8:40_1_8_1"><span class="lab"><font color="Green" title="E48">A143</font></span></a>, <a class="txt" href="#E6:40_1_8_1"><span class="lab"><font color="Green" title="E47">A142</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E12:40_1_8_1"/><span class="lab"><font color="Green" title="E51">A146</font></span>: 
<font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_8_1"><span class="lab"><font color="Green" title="E45">A140</font></span></a>, <a class="txt" href="#E10:40_1_8_1"><span class="lab"><font color="Green" title="E49">A144</font></span></a>, <a class="txt" href="#E8:40_1_8_1"><span class="lab"><font color="Green" title="E48">A143</font></span></a>, <a class="txt" href="#E6:40_1_8_1"><span class="lab"><font color="Green" title="E47">A142</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E13:40_1_8_1"/>
(  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1_8_1_1"/><span class="lab"><font color="Green" title="E52">A147</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:40_1_8_1_1_1"/><span class="lab"><font color="Green" title="E53">A148</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">k</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font> implies  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> )</span><br/><a NAME="E1:40_1_8_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">f</font> &amp; <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c14">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c14">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E2:40_1_8_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">k</font> implies  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_8_1_1_1"><span class="lab"><font color="Green" title="E53">A148</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:40_1_8_1_1_1"/><span class="kw">then </span>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c14">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E1:40_1_8_1_1"><span class="lab"><font color="Green" title="E52">A147</font></span></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_8_1_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c14">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E4:40_1_8_1"><span class="lab"><font color="Green" title="E46">A141</font></span></a>, <a class="txt" href="#E12:40_1_8_1"><span class="lab"><font color="Green" title="E51">A146</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:40_1_8_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E14:40_1_8_1"/><span class="lab"><font color="Green" title="E52">A149</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E15:40_1_8_1"/><span class="lab"><font color="Green" title="E53">A150</font></span>: 
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E17:40_1_8_1"/><span class="lab"><font color="Green" title="E54">A151</font></span>: 
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c14">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E15:40_1_8_1"><span class="lab"><font color="Green" title="E53">A150</font></span></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></span>;<br/>
<a NAME="E18:40_1_8_1"/><span class="lab"><font color="Green" title="E55">A152</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">ksi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:40_1"><span class="lab"><font color="Green" title="E43">A138</font></span></a></span>;<br/>
<a NAME="E19:40_1_8_1"/><span class="lab"><font color="Green" title="E56">A153</font></span>: 
<font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">h</font> = <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">a1</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><br/><a NAME="E22:40_1_8_1"/><span class="lab"><font color="Green" title="E57">A154</font></span>: 
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">rho</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">h</font>
 <span class="kw">and </span><br/><a NAME="E23:40_1_8_1"/><span class="lab"><font color="Green" title="E58">A155</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; (  ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b2">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) )
 <span class="kw">and </span><br/><a NAME="E24:40_1_8_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st  ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> st <br/>( <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; (  ex <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font> st <br/>( <font color="Olive" title="b3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">b</font> &amp;  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) or ( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal4.html#K2" title="ORDINAL4:func.2">0-element_of</a> <font color="Maroon" title="c1">W</font> &amp; ( for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>  holds not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b2">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Olive" title="b3">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span> ) ) ) ) holds <br/><font color="Maroon" title="c16">a1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a></span>;<br/>
<a NAME="E25:40_1_8_1"/><span class="lab"><font color="Green" title="E59">A156</font></span>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></span>;<br/>
<a NAME="E26:40_1_8_1"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c14">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40_1_8_1"><span class="lab"><font color="Green" title="E54">A151</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font><span class="kw"> such that </span><br/><a NAME="E28:40_1_8_1"/><span class="lab"><font color="Green" title="E60">A157</font></span>: 
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font>
 <span class="kw">and </span><br/><a NAME="E29:40_1_8_1"/><span class="lab"><font color="Green" title="E61">A158</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40_1_8_1"><span class="lab"><font color="Green" title="E58">A155</font></span></a></span>;<br/>
<a NAME="E30:40_1_8_1"/>
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E31:40_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rho</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E22:40_1_8_1"><span class="lab"><font color="Green" title="E57">A154</font></span></a>, <a class="txt" href="#E25:40_1_8_1"><span class="lab"><font color="Green" title="E59">A156</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E32:40_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">si</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1_8_1"><span class="lab"><font color="Green" title="E56">A153</font></span></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></span>;<br/>
<a NAME="E33:40_1_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c16">a1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E11:40_1_8_1"><span class="lab"><font color="Green" title="E50">A145</font></span></a>, <a class="txt" href="#E18:40_1_8_1"><span class="lab"><font color="Green" title="E55">A152</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">m9</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40_1_8_1"><span class="lab"><font color="Green" title="E60">A157</font></span></a></span>;<br/>
<a NAME="E35:40_1_8_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40_1_8_1"><span class="lab"><font color="Green" title="E59">A156</font></span></a>, <a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></span>;<br/>
<a NAME="E36:40_1_8_1"/><span class="kw">then </span>
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m9</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:40_1_8_1"><span class="lab"><font color="Green" title="E61">A158</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>;<br/>
<a NAME="E37:40_1_8_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">f</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c18">m9</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E4:40_1_8_1"><span class="lab"><font color="Green" title="E46">A141</font></span></a>, <a class="txt" href="#E12:40_1_8_1"><span class="lab"><font color="Green" title="E51">A146</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:40_1_8_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E14:40_1_8_1"><span class="lab"><font color="Green" title="E52">A149</font></span></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>

<a NAME="E39:40_1_8_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_8_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:40_1_8_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:40_1_8_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1_8_1_2"/><span class="lab"><font color="Green" title="E62">A159</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:40_1_8_1_2"/><span class="lab"><font color="Green" title="E63">A160</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c20">y</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_8_1_2"><span class="lab"><font color="Green" title="E62">A159</font></span></a></span>;<br/>
<a NAME="E7:40_1_8_1_2"/>
( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> or <font color="Maroon" title="c21">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> )
 
;<br/>
<a NAME="E8:40_1_8_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m9</font> or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40_1_8_1"><span class="lab"><font color="Green" title="E59">A156</font></span></a>, <a class="txt" href="#E5:40_1_8_1_2"><span class="lab"><font color="Green" title="E63">A160</font></span></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a>, <a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:40_1_8_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c19">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c13">f</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font></span>)</span>,<font color="Maroon" title="c17">m</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_9"/><span class="lab"><font color="Green" title="E45">A161</font></span>: 
not  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]</span><br/><span class="kw">thus </span><a NAME="E2:40_1_9"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c10">phi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c10">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:40_1_9_1"/>
( <font color="Maroon" title="c10">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c10">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E17:40_1"><span class="lab"><font color="Green" title="E34">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">a</font> implies for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:40_1_9_1"/><span class="lab"><font color="Green" title="E46">A162</font></span>: 
( <font color="Maroon" title="c10">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:40_1_9_1"/>
(  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_9_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:40_1_9_1_1"/><span class="lab"><font color="Green" title="E47">A163</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 
;<br/>
<span class="kw">assume </span><a NAME="E2:40_1_9_1_1"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<a NAME="E3:40_1_9_1_1"/><span class="kw">then </span>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E1:40_1_9_1_1"><span class="lab"><font color="Green" title="E47">A163</font></span></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></span>;<br/>
<a NAME="E4:40_1_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E2:40_1_9_1"><span class="lab"><font color="Green" title="E46">A162</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:40_1_9_1_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E1:40_1_9"><span class="lab"><font color="Green" title="E45">A161</font></span></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E4:40_1_9_1"/><span class="lab"><font color="Green" title="E47">A164</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 
;<br/>
<span class="kw">assume </span><a NAME="E5:40_1_9_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<a NAME="E6:40_1_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E4:40_1_9_1"><span class="lab"><font color="Green" title="E47">A164</font></span></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></span>;<br/>
<a NAME="E7:40_1_9_1"/><span class="kw">then </span>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E2:40_1_9_1"><span class="lab"><font color="Green" title="E46">A162</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40_1_9_1"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c12">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E1:40_1_9"><span class="lab"><font color="Green" title="E45">A161</font></span></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E29:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:40_1"><span class="lab"><font color="Green" title="E44">A139</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:40"/><span class="lab"><font color="Green" title="E23">A165</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Olive" title="b1">H</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_2"/>
<font color="Maroon" title="c4">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font>] or  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<a NAME="E2:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A166</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T58" title="ZF_LANG:th.58">ZF_LANG:58</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c5">fi1</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><a NAME="E4:40_2"/><span class="lab"><font color="Green" title="E25">A167</font></span>: 
<font color="Maroon" title="c5">fi1</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E5:40_2"/><span class="lab"><font color="Green" title="E26">A168</font></span>: 
<font color="Maroon" title="c5">fi1</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E6:40_2"/><span class="lab"><font color="Green" title="E27">A169</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">fi2</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><a NAME="E8:40_2"/><span class="lab"><font color="Green" title="E28">A170</font></span>: 
<font color="Maroon" title="c6">fi2</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E9:40_2"/><span class="lab"><font color="Green" title="E29">A171</font></span>: 
<font color="Maroon" title="c6">fi2</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E10:40_2"/><span class="lab"><font color="Green" title="E30">A172</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]</span><br/>

<span class="kw">take </span><font color="Maroon" title="c7">phi</font> = <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c5">fi1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c7">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) ) )</span><br/>

<a NAME="E11:40_2"/>
 ex <font color="Olive" title="b1">fi</font> being  <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> st <br/>( <font color="Olive" title="b1">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K5" title="ORDINAL4:func.5">*</a> <font color="Maroon" title="c5">fi1</font> &amp; <font color="Olive" title="b1">fi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2"><span class="lab"><font color="Green" title="E25">A167</font></span></a>, <a class="txt" href="#E8:40_2"><span class="lab"><font color="Green" title="E28">A170</font></span></a>, <a class="ref" href="ordinal4.html#T13" title="ORDINAL4:th.13">ORDINAL4:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:40_2"/>
( <font color="Maroon" title="c7">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c7">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2"><span class="lab"><font color="Green" title="E25">A167</font></span></a>, <a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E26">A168</font></span></a>, <a class="txt" href="#E9:40_2"><span class="lab"><font color="Green" title="E29">A171</font></span></a>, <a class="ref" href="ordinal4.html#T17" title="ORDINAL4:th.17">ORDINAL4:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font> implies for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:40_2"/><span class="lab"><font color="Green" title="E31">A173</font></span>: 
<font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E14:40_2"/><span class="lab"><font color="Green" title="E32">A174</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<a NAME="E15:40_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">fi1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<a NAME="E16:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A175</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2"><span class="lab"><font color="Green" title="E25">A167</font></span></a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<a NAME="E17:40_2"/><span class="lab"><font color="Green" title="E34">A176</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<a NAME="E18:40_2"/><span class="lab"><font color="Green" title="E35">A177</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">fi2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E36">A178</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_2_1"/>
<font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:40_2"><span class="lab"><font color="Green" title="E33">A175</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E3:40_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A179</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/><a NAME="E4:40_2_1"/><span class="lab"><font color="Green" title="E38">A180</font></span>: 
<font color="Maroon" title="c7">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40_2"><span class="lab"><font color="Green" title="E34">A176</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/><a NAME="E5:40_2_1"/>
<font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">fi2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal4.html#T36" title="ORDINAL4:th.36">ORDINAL4:36</a></span>;<br/><a NAME="E6:40_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fi1</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_2"><span class="lab"><font color="Green" title="E28">A170</font></span></a>, <a class="txt" href="#E3:40_2_1"><span class="lab"><font color="Green" title="E37">A179</font></span></a>, <a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a></span>;<br/><span class="kw">hence </span><a NAME="E7:40_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_2"><span class="lab"><font color="Green" title="E28">A170</font></span></a>, <a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E31">A173</font></span></a>, <a class="txt" href="#E18:40_2"><span class="lab"><font color="Green" title="E35">A177</font></span></a>, <a class="txt" href="#E4:40_2_1"><span class="lab"><font color="Green" title="E38">A180</font></span></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a>, <a class="ref" href="ordinal4.html#T10" title="ORDINAL4:th.10">ORDINAL4:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E20:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A181</font></span>: 
<font color="Maroon" title="c6">fi2</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E31">A173</font></span></a>, <a class="txt" href="#E17:40_2"><span class="lab"><font color="Green" title="E34">A176</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<span class="kw">thus </span><a NAME="E21:40_2"/>
(  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_2_2"/><span class="lab"><font color="Green" title="E38">A182</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<a NAME="E2:40_2_2"/><span class="kw">then </span>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E24">A166</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>;<br/>
<a NAME="E3:40_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A183</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_2"><span class="lab"><font color="Green" title="E30">A172</font></span></a>, <a class="txt" href="#E14:40_2"><span class="lab"><font color="Green" title="E32">A174</font></span></a>, <a class="txt" href="#E20:40_2"><span class="lab"><font color="Green" title="E37">A181</font></span></a></span>;<br/>
<a NAME="E4:40_2_2"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E24">A166</font></span></a>, <a class="txt" href="#E1:40_2_2"><span class="lab"><font color="Green" title="E38">A182</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>;<br/>
<a NAME="E5:40_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_2"><span class="lab"><font color="Green" title="E27">A169</font></span></a>, <a class="txt" href="#E14:40_2"><span class="lab"><font color="Green" title="E32">A174</font></span></a>, <a class="txt" href="#E19:40_2"><span class="lab"><font color="Green" title="E36">A178</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_2_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E24">A166</font></span></a>, <a class="txt" href="#E3:40_2_2"><span class="lab"><font color="Green" title="E39">A183</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E22:40_2"/><span class="lab"><font color="Green" title="E38">A184</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<a NAME="E23:40_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E24">A166</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>;<br/>
<a NAME="E24:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A185</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_2"><span class="lab"><font color="Green" title="E30">A172</font></span></a>, <a class="txt" href="#E14:40_2"><span class="lab"><font color="Green" title="E32">A174</font></span></a>, <a class="txt" href="#E20:40_2"><span class="lab"><font color="Green" title="E37">A181</font></span></a></span>;<br/>
<a NAME="E25:40_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E24">A166</font></span></a>, <a class="txt" href="#E22:40_2"><span class="lab"><font color="Green" title="E38">A184</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>;<br/>
<a NAME="E26:40_2"/><span class="kw">then </span>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_2"><span class="lab"><font color="Green" title="E27">A169</font></span></a>, <a class="txt" href="#E14:40_2"><span class="lab"><font color="Green" title="E32">A174</font></span></a>, <a class="txt" href="#E19:40_2"><span class="lab"><font color="Green" title="E36">A178</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:40_2"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E24">A166</font></span></a>, <a class="txt" href="#E24:40_2"><span class="lab"><font color="Green" title="E39">A185</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:40"/><span class="lab"><font color="Green" title="E24">A186</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:40_3"/><span class="lab"><font color="Green" title="E25">A187</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">phi</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></span>;<br/>
<a NAME="E3:40_3"/><span class="lab"><font color="Green" title="E26">A188</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c1">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">phi</font> = <font color="Maroon" title="c4">phi</font> as   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#D8" title="ORDINAL2:def.8">ORDINAL2:def 8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">phi</font> = <font color="Maroon" title="c5">phi</font> as   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3"><span class="lab"><font color="Green" title="E25">A187</font></span></a>, <a class="txt" href="#E3:40_3"><span class="lab"><font color="Green" title="E26">A188</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E6:40_3"/><span class="lab"><font color="Green" title="E27">A189</font></span>: 
( <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  or <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D15" title="ZF_LANG:def.15">ZF_LANG:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">H</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">phi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:40_3"/><span class="lab"><font color="Green" title="E28">A190</font></span>: 
<font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal2.html#D16" title="ORDINAL2:def.16">ORDINAL2:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">phi</font> or <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> or  not <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">phi</font> or <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:40_3_1"/><span class="lab"><font color="Green" title="E29">A191</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">phi</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font></span><br/>

<a NAME="E2:40_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:40_3_1"/>
<font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_1"><span class="lab"><font color="Green" title="E29">A191</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E8:40_3"/>
<font color="Maroon" title="c6">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ordinal2.html#D17" title="ORDINAL2:def.17">ORDINAL2:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">phi</font> or <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c8">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> or <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">phi</font> or <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c8">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  or  not <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> or <font color="Maroon" title="c9">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_3_2"/><span class="lab"><font color="Green" title="E29">A192</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">phi</font>
 <span class="kw">and </span><br/><a NAME="E2:40_3_2"/><span class="lab"><font color="Green" title="E30">A193</font></span>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">A</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )
 <span class="kw">and </span><br/><a NAME="E3:40_3_2"/><span class="lab"><font color="Green" title="E31">A194</font></span>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span><br/>

<a NAME="E4:40_3_2"/><span class="lab"><font color="Green" title="E32">A195</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_2"><span class="lab"><font color="Green" title="E29">A192</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font> = 
<font color="Maroon" title="c6">phi</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></span>
<br/>.= 
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">phi</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3"><span class="lab"><font color="Green" title="E25">A187</font></span></a>, <a class="ref" href="funct_1.html#T43" title="FUNCT_1:th.43">FUNCT_1:43</a></span>
<br/>.= 
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c8">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_3_2"><span class="lab"><font color="Green" title="E32">A195</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E6:40_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A196</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<a NAME="E7:40_3_2"/>
<font color="Maroon" title="c6">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_2"><span class="lab"><font color="Green" title="E29">A192</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<a NAME="E8:40_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A197</font></span>: 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_3_2"><span class="lab"><font color="Green" title="E31">A194</font></span></a>, <a class="txt" href="#E6:40_3_2"><span class="lab"><font color="Green" title="E33">A196</font></span></a>, <a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></span>;<br/>
<a NAME="E9:40_3_2"/><span class="lab"><font color="Green" title="E35">A198</font></span>: 
<font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_3"><span class="lab"><font color="Green" title="E28">A190</font></span></a>, <a class="ref" href="ordinal4.html#T15" title="ORDINAL4:th.15">ORDINAL4:15</a></span>;<br/>
<a NAME="E10:40_3_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_3_2"><span class="lab"><font color="Green" title="E32">A195</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:40_3_2"/>
<font color="Maroon" title="c9">B</font> <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c6">phi</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_3_2"><span class="lab"><font color="Green" title="E30">A193</font></span></a>, <a class="txt" href="#E8:40_3_2"><span class="lab"><font color="Green" title="E34">A197</font></span></a>, <a class="txt" href="#E9:40_3_2"><span class="lab"><font color="Green" title="E35">A198</font></span></a>, <a class="ref" href="ordinal4.html#T8" title="ORDINAL4:th.8">ORDINAL4:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font> implies for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:40_3"/>
<font color="Maroon" title="c6">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E10:40_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><br/>

<span class="kw">thus </span><a NAME="E11:40_3"/>
(  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_3_3"/><span class="lab"><font color="Green" title="E29">A199</font></span>: 
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/>

<a NAME="E2:40_3_3"/><span class="lab"><font color="Green" title="E30">A200</font></span>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">A201</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_3_3_1"/>
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/><a NAME="E2:40_3_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A202</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T54" title="ZF_LANG:th.54">ZF_LANG:54</a></span>;<br/><a NAME="E3:40_3_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <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="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_3"><span class="lab"><font color="Green" title="E29">A199</font></span></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_3_3_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_3_3"><span class="lab"><font color="Green" title="E30">A200</font></span></a>, <a class="txt" href="#E2:40_3_3_1"><span class="lab"><font color="Green" title="E32">A202</font></span></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_3_3_2"/>
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/><a NAME="E2:40_3_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A203</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a></span>;<br/><a NAME="E3:40_3_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_3_3"><span class="lab"><font color="Green" title="E29">A199</font></span></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_3_3_2"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_3_3"><span class="lab"><font color="Green" title="E30">A200</font></span></a>, <a class="txt" href="#E2:40_3_3_2"><span class="lab"><font color="Green" title="E32">A203</font></span></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:40_3_3"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_3"><span class="lab"><font color="Green" title="E27">A189</font></span></a>, <a class="txt" href="#E3:40_3_3"><span class="lab"><font color="Green" title="E31">A201</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E12:40_3"/><span class="lab"><font color="Green" title="E29">A204</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/>

<a NAME="E13:40_3"/><span class="lab"><font color="Green" title="E30">A205</font></span>: 
<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_refle.html#T24" target="_self" title="ZF_REFLE:th.24">Th24</a>, <a class="ref" href="zf_lang1.html#D2" title="ZF_LANG1:def.2">ZF_LANG1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">A206</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_3_4"/>
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/><a NAME="E2:40_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A207</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T54" title="ZF_LANG:th.54">ZF_LANG:54</a></span>;<br/><a NAME="E3:40_3_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_3"><span class="lab"><font color="Green" title="E29">A204</font></span></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_3_4"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_3"><span class="lab"><font color="Green" title="E30">A205</font></span></a>, <a class="txt" href="#E2:40_3_4"><span class="lab"><font color="Green" title="E32">A207</font></span></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_3_5"/>
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/><a NAME="E2:40_3_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A208</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a></span>;<br/><a NAME="E3:40_3_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:40_3"><span class="lab"><font color="Green" title="E29">A204</font></span></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_3_5"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_3"><span class="lab"><font color="Green" title="E30">A205</font></span></a>, <a class="txt" href="#E2:40_3_5"><span class="lab"><font color="Green" title="E32">A208</font></span></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:40_3"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c9">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_3"><span class="lab"><font color="Green" title="E27">A189</font></span></a>, <a class="txt" href="#E14:40_3"><span class="lab"><font color="Green" title="E31">A206</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:40"/><span class="lab"><font color="Green" title="E25">A209</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_4"/>
<font color="Maroon" title="c4">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] )</span><br/>

<a NAME="E2:40_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A210</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c5">phi</font> being   <a href="ordinal4.html#NM2" title="ORDINAL4:NM.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><span class="kw"> such that </span><a NAME="E4:40_4"/><span class="lab"><font color="Green" title="E27">A211</font></span>: 
( <font color="Maroon" title="c5">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c5">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">and </span><br/><a NAME="E5:40_4"/><span class="lab"><font color="Green" title="E28">A212</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">phi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c5">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:40_4"/>
( <font color="Maroon" title="c5">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>  &amp; <font color="Maroon" title="c5">phi</font> is  <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_4"><span class="lab"><font color="Green" title="E27">A211</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>  st <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> implies for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E7:40_4"/><span class="lab"><font color="Green" title="E29">A213</font></span>: 
( <font color="Maroon" title="c5">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font></span>)</span> holds <br/> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> iff <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:40_4"/>
(  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> implies  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_4_1"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<a NAME="E2:40_4_1"/><span class="kw">then </span>
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_4"><span class="lab"><font color="Green" title="E26">A210</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>;<br/>
<a NAME="E3:40_4_1"/><span class="kw">then </span>
not <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_4"><span class="lab"><font color="Green" title="E28">A212</font></span></a>, <a class="txt" href="#E7:40_4"><span class="lab"><font color="Green" title="E29">A213</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40_4_1"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_4"><span class="lab"><font color="Green" title="E26">A210</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E9:40_4"/>
<font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font></span><br/>

<a NAME="E10:40_4"/><span class="kw">then </span>
not <font color="Maroon" title="c2">L</font> <a href="zf_refle.html#K5" title="ZF_REFLE:func.5">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_4"><span class="lab"><font color="Green" title="E26">A210</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>;<br/>
<a NAME="E11:40_4"/><span class="kw">then </span>
not  <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_4"><span class="lab"><font color="Green" title="E28">A212</font></span></a>, <a class="txt" href="#E7:40_4"><span class="lab"><font color="Green" title="E29">A213</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:40_4"/>
 <a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4" title="ZF_REFLE:func.4">Union</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">!</a> <font color="Maroon" title="c7">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_4"><span class="lab"><font color="Green" title="E26">A210</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E10:40"/>
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_lang.html#S1" title="ZF_LANG:sch.1">ZF_LANG:sch 1</a>(<a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E24">A186</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E25">A209</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E23">A165</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E22">A6</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
