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<span class="kw">let </span><font color="Maroon" title="c1">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">R</font> being   <a href="fomodel4.html#NM1" title="FOMODEL4:NM.1">Rule</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">Seqts</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span>  st <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">Seqts</font> holds <br/><font color="Maroon" title="c1">Y</font> is <font color="Olive" title="b3">Seqts</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">R</font> being   <a href="fomodel4.html#NM1" title="FOMODEL4:NM.1">Rule</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">Seqts</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span>  st <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">Seqts</font> holds <br/><font color="Maroon" title="c1">Y</font> is <font color="Olive" title="b2">Seqts</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="fomodel4.html#NM1" title="FOMODEL4:NM.1">Rule</a> of <font color="Maroon" title="c2">S</font>; <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">Seqts</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span>  st <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">Seqts</font> holds <br/><font color="Maroon" title="c1">Y</font> is <font color="Olive" title="b1">Seqts</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">Seqts</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span>; <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="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font> implies <font color="Maroon" title="c1">Y</font> is <font color="Maroon" title="c4">Seqts</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a>  )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">D</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</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="c6">RR</font> = <span class="p1">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> ;<br/>
<a NAME="E1:67"/>
( <font color="Maroon" title="c4">Seqts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></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">then </span><span class="lab"><font color="Green" title="E32">B0</font></span>: <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> (<font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c4">Seqts</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>
<br/>.= 
<font color="Maroon" title="c3">R</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">Seqts</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>

;<br/>
<a NAME="E3:67"/>
 <a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#T18" title="LANG1:th.18">LANG1:18</a></span>;<br/>
<a NAME="E4:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B1</font></span>: 
<font color="Maroon" title="c3">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E24"><span class="lab"><font color="Green" title="E19">Lm8</font></span></a></span>;<br/>
<a NAME="E5:67"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">Seqts</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="#E2:67"><span class="lab"><font color="Green" title="E32">B0</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E33">B1</font></span></a>, <a class="ref" href="relat_1.html#T124" title="RELAT_1:th.124">RELAT_1:124</a></span>;<br/>
<a NAME="E6:67"/><span class="kw">then </span>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</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"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font></span>)</span></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="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">Seqts</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="zfmisc_1.html#T77" title="ZFMISC_1:th.77">ZFMISC_1:77</a></span>;<br/>
<a NAME="E7:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B3</font></span>: 
<font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">Seqts</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="zfmisc_1.html#T25" title="ZFMISC_1:th.25">ZFMISC_1:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:67"/>
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font>
 ; <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="c1">Y</font> is <font color="Maroon" title="c4">Seqts</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> </span><br/>

<a NAME="E9:67"/><span class="kw">then </span>
<font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="lang1.html#K13" title="LANG1:func.13">[*]</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">Seqts</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="txt" href="#E7:67"><span class="lab"><font color="Green" title="E34">B3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:67"/>
<font color="Maroon" title="c1">Y</font> is <font color="Maroon" title="c4">Seqts</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">R</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D6" target="_self" title="FOMODEL4:def.6">DefDerivable</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>
