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

<span class="kw">set </span><font color="Maroon" title="c3">L1</font> = <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">a1</font> =  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">SS1</font> =  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">L2</font> = <font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">SS2</font> =  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">a2</font> =  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">E1</font> =  <a href="fomodel1.html#K7" title="FOMODEL1:func.7">TheEqSymbOf</a> <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">N1</font> =  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">E2</font> =  <a href="fomodel1.html#K7" title="FOMODEL1:func.7">TheEqSymbOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">N2</font> =  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">Y</font> = <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E39">B0</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c14">Y</font>

;<br/>
<a NAME="E3:142_1_1"/>
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D42" target="_self" title="FOMODEL1:def.42">DefExt1</a></span>;<br/>
<a NAME="E4:142_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">B1</font></span>: <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c14">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_1_1"><span class="lab"><font color="Green" title="E39">B0</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c14">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">NN1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">NN2</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E8:142_1_1"/>
( <font color="Maroon" title="c15">NN1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font> &amp; <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">NN11</font> = <font color="Maroon" title="c15">NN1</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E10:142_1_1"/>
(  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">NN2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">NN2</font></span>)</span> &amp;  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">NN1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">NN1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<span class="kw">then </span> <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">NN2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">NN1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:142_1_1"><span class="lab"><font color="Green" title="E40">B1</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">NN1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">NN1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D41" target="_self" title="FOMODEL1:def.41">DefExt0</a></span>
<br/>.= 
<font color="Maroon" title="c15">NN1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">NN1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>
<br/>.= 
<font color="Maroon" title="c17">NN11</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c17">NN11</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:142_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="fomodel1.html#K40" title="FOMODEL1:func.40">extendWith</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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