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<span class="kw">set </span><font color="Maroon" title="c4">Q</font> = <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">O</font> =  <a href="fomodel4.html#K2" title="FOMODEL4:func.2">OneStep</a> <font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">F</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b1">mm</font>,<font color="Maroon" title="c2">D</font>) <a href="fomodel4.html#K3" title="FOMODEL4:func.3">-derivables</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">X</font></span>)</span> where <font color="Olive" title="b1">mm</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">IT</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="c7">IT</font> is <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a>  implies <font color="Maroon" title="c7">IT</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V2" title="FOMODEL4:attr.2">-sequents-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:142_1_1"/>
<font color="Maroon" title="c7">IT</font> is <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</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="c7">IT</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V2" title="FOMODEL4:attr.2">-sequents-like</a> </span><br/>

<a NAME="E2:142_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">IT</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> <font color="Maroon" title="c2">D</font></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">X</font></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="fomodel4.html#D6" target="_self" title="FOMODEL4:def.6">DefDerivable</a></span>;<br/>
<a NAME="E3:142_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">B0</font></span>: 
<font color="Maroon" title="c7">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b1">mm</font>,<font color="Maroon" title="c2">D</font>) <a href="fomodel4.html#K3" title="FOMODEL4:func.3">-derivables</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">X</font></span>)</span> where <font color="Olive" title="b1">mm</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E27"><span class="lab"><font color="Green" title="E22">Lm18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:142_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">IT</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:142_1_1_1"/><span class="lab"><font color="Green" title="E95">C0</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Olive" title="b1">mm</font>,<font color="Maroon" title="c2">D</font>) <a href="fomodel4.html#K3" title="FOMODEL4:func.3">-derivables</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">X</font></span>)</span> where <font color="Olive" title="b1">mm</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:142_1_1"><span class="lab"><font color="Green" title="E94">B0</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">mm</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:142_1_1_1"/><span class="lab"><font color="Green" title="E96">C1</font></span>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c10">mm</font>,<font color="Maroon" title="c2">D</font>) <a href="fomodel4.html#K3" title="FOMODEL4:func.3">-derivables</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:142_1_1_1"><span class="lab"><font color="Green" title="E95">C0</font></span></a></span>;<br/><a NAME="E6:142_1_1_1"/>
<font color="Maroon" title="c8">x</font> is <font color="Maroon" title="c10">mm</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D7" target="_self" title="FOMODEL4:def.7">DefDerivable3</a>, <a class="txt" href="#E3:142_1_1_1"><span class="lab"><font color="Green" title="E95">C0</font></span></a>, <a class="txt" href="#E5:142_1_1_1"><span class="lab"><font color="Green" title="E96">C1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:142_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D2" target="_self" title="FOMODEL4:def.2">DefSeqtLike</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>
<a NAME="E5:142_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</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/>
<span class="kw">hence </span><a NAME="E6:142_1_1"/>
<font color="Maroon" title="c7">IT</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V2" title="FOMODEL4:attr.2">-sequents-like</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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