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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">  for <font color="Olive" title="b1">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c1">S</font> holds   <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <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="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> is <font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">AF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> is  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  implies <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:73_1"/>
<font color="Maroon" title="c4">x</font> is  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c4">x</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E41"><span class="lab"><font color="Green" title="E36">Lm33</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:73_1"/><span class="lab"><font color="Green" title="E38">C0</font></span>: 
<font color="Maroon" title="c5">xx</font> is <font color="Maroon" title="c6">m</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D25" title="FOMODEL2:def.25">FOMODEL2:def 25</a></span>;<br/><a NAME="E5:73_1"/>
<font color="Maroon" title="c5">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E38">C0</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:73_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>
 <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>; <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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:73"/>
 <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> is <font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D16" target="_self" title="FOMODEL4:def.16">DefExpanded</a></span>; <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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