<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp; <span class="p1">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">o</font> =  the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">z</font> =  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">O</font> =  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">SS</font> =  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">e</font> =  <a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">n</font> =  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">A</font> =  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">D</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">L</font> =  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">T</font> =  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">OP</font> =  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">B</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E1:26"/><span class="lab"><font color="Green" title="E15">B3</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> = 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E15">B3</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T136" title="RELAT_1:th.136">RELAT_1:136</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp; <span class="p1">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font></span>)</span> = 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></span>
<br/>.= 
 <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font>
<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>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<a NAME="E4:26"/><span class="lab"><font color="Green" title="E16">Z0</font></span>: 
 <a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D17" target="_self" title="FOMODEL1:def.17">DefRelational</a></span>;<br/>
 <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> = 
<span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T52" title="XBOOLE_1:th.52">XBOOLE_1:52</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <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#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>

<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E16">Z0</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:26"/>
<span class="p1">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:26"/>
 <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a>, <a class="ref" href="zfmisc_1.html#T7" title="ZFMISC_1:th.7">ZFMISC_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<a NAME="E9:26"/>
(  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> &amp;  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T69" title="FUNCT_1:th.69">FUNCT_1:69</a>, <a class="ref" href="relat_1.html#T143" title="RELAT_1:th.143">RELAT_1:143</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:26"/>
 <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<a NAME="E11:26"/>
 <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"> the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T69" title="FUNCT_1:th.69">FUNCT_1:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:26"/>
 <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<span class="kw">thus </span><a NAME="E13:26"/>
 <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T143" title="RELAT_1:th.143">RELAT_1:143</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2 = 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D23" target="_self" title="FOMODEL1:def.23">DefEligible</a></span>
<br/>.= 
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font>

;<br/>
<a NAME="E15:26"/><span class="kw">then </span>
 not  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a>  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E16:26"/><span class="kw">then </span>
 not  the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E17:26"/><span class="kw">then </span>
(  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a NAME="E18:26"/><span class="kw">then </span>
 the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></span>;<br/>
<a NAME="E19:26"/><span class="kw">then </span>
 <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:26"/>
 <a href="fomodel1.html#K5" title="FOMODEL1:func.5">TermSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K9" title="FOMODEL1:func.9">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<a NAME="E21:26"/>
 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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<a NAME="E22:26"/><span class="kw">then </span>
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E23:26"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:26"/>
 <a href="fomodel1.html#K3" title="FOMODEL1:func.3">OpSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T143" title="RELAT_1:th.143">RELAT_1:143</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font></span><br/>

<span class="kw">thus </span><a NAME="E25:26"/>
 <a href="fomodel1.html#K6" title="FOMODEL1:func.6">LowerCompoundersOf</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel1.html#K17" title="FOMODEL1:func.17">AtomicFormulaSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
