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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">C</font> being   <a href="orders_4.html#V1" title="ORDERS_4:attr.1">countable</a>  <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">phi</font> being   <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="Olive" title="b1">C</font>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  holds <br/><font color="Olive" title="b2">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel4.html#V18" title="FOMODEL4:attr.18">-provable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="orders_4.html#V1" title="ORDERS_4:attr.1">countable</a>  <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">phi</font> being   <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="c2">C</font>  st <font color="Maroon" title="c1">X</font> <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="c2">C</font> &amp; <font color="Olive" title="b1">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  holds <br/><font color="Olive" title="b1">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel4.html#V18" title="FOMODEL4:attr.18">-provable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">phi</font> be   <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="c2">C</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">X</font> <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="c2">C</font> &amp; <font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  implies <font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel4.html#V18" title="FOMODEL4:attr.18">-provable</a>  )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c2">C</font> as   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">DD</font> = <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="fomodel4.html#K34" title="FOMODEL4:func.34">R9</a> <font color="Maroon" title="c4">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c4">S</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">FF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">D</font> = <font color="Maroon" title="c2">C</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a> ;<br/>
<span class="kw">assume </span><a NAME="E3:263"/>
<font color="Maroon" title="c1">X</font> <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="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  or <font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel4.html#V18" title="FOMODEL4:attr.18">-provable</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c1">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">C</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E5:263"/>
<font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</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="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel4.html#V18" title="FOMODEL4:attr.18">-provable</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">phii</font> = <font color="Maroon" title="c3">phi</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <font color="Maroon" title="c1">X</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">C</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">psi</font> =  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c9">phii</font></span>)</span>;<br/>
<a NAME="E7:263"/>
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c9">phii</font></span>)</span> is <font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c2">C</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E173"><span class="lab"><font color="Green" title="E140">Lm55</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">H</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c12">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="E9:263"/><span class="lab"><font color="Green" title="E141">B0</font></span>: 
( <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">H</font>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c9">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is <font color="Maroon" title="c12">m</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">C</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <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#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">seqt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">H</font>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c9">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as  <font color="Maroon" title="c2">C</font> <a href="fomodel4.html#V1" title="FOMODEL4:attr.1">-sequent-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:263"><span class="lab"><font color="Green" title="E141">B0</font></span></a></span>;<br/>
<a NAME="E11:263"/><span class="lab"><font color="Green" title="E142">BB1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">seqt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c11">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">HH</font> = <font color="Maroon" title="c11">H</font> as  <font color="Maroon" title="c4">S</font> <a href="fomodel4.html#V13" title="FOMODEL4:attr.13">-premises-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:263"><span class="lab"><font color="Green" title="E142">BB1</font></span></a>, <a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">HC</font> = <font color="Maroon" title="c11">H</font> as  <font color="Maroon" title="c2">C</font> <a href="fomodel4.html#V13" title="FOMODEL4:attr.13">-premises-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:263"><span class="lab"><font color="Green" title="E142">BB1</font></span></a>, <a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c3">phi</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="c4">S</font> ;<br/>
<a NAME="E15:263"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">HC</font>,<font color="Maroon" title="c3">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> 1 is 1,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">HC</font>,<span class="p3">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p4">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c3">phi</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><a href="fomodel4.html#K34" title="FOMODEL4:func.34">R9</a> <font color="Maroon" title="c2">C</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 
;<br/>
<a NAME="E16:263"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">HC</font>,<font color="Maroon" title="c3">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is <font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="fomodel4.html#K34" title="FOMODEL4:func.34">R9</a> <font color="Maroon" title="c2">C</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E38"><span class="lab"><font color="Green" title="E33">Lm28</font></span></a>, <a class="txt" href="#E9:263"><span class="lab"><font color="Green" title="E141">B0</font></span></a></span>;<br/>
<a NAME="E17:263"/><span class="kw">then </span>
<font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c8">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="fomodel4.html#K34" title="FOMODEL4:func.34">R9</a> <font color="Maroon" title="c2">C</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:263"><span class="lab"><font color="Green" title="E141">B0</font></span></a>, <a class="ref" href="fomodel4.html#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:263"/>
<font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">X</font> <a href="fomodel4.html#V18" title="FOMODEL4:attr.18">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D70" target="_self" title="FOMODEL4:def.70">DefProvable4</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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