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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">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">S</font>  st  <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Olive" title="b2">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  not <font color="Olive" title="b2">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  holds <br/><font color="Olive" title="b2">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Olive" title="b1">S</font></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">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">S</font>  st  <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  not <font color="Olive" title="b1">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  holds <br/><font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font></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">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#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  not <font color="Maroon" title="c3">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  implies <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">p</font> =  <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c3">phi</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Phim</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">E</font> = <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K36" title="FOMODEL2:func.36">-ExFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">PhiM</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">N</font> = <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="lab"><font color="Green" title="E60">Z0</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K36" title="FOMODEL2:func.36">-ExFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T9" target="_self" title="FOMODEL2:th.9">Lm37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K36" title="FOMODEL2:func.36">-ExFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></span>
;<br/>
<a NAME="E2:136"/><span class="lab"><font color="Green" title="E61">B0</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:136"/>
 <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  or <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<a NAME="E4:136"/><span class="kw">then </span>
( <font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  &amp;  not <font color="Maroon" title="c3">phi</font> is <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E61">B0</font></span></a>, <a class="ref" href="fomodel2.html#D31" target="_self" title="FOMODEL2:def.31">DefDepth</a></span>;<br/>
<a NAME="E5:136"/><span class="kw">then </span>
( <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  not <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D24" target="_self" title="FOMODEL2:def.24">DefWff1</a></span>;<br/>
<a NAME="E6:136"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">Z1</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:136"/>
 not <font color="Maroon" title="c3">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E8:136"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K36" title="FOMODEL2:func.36">-ExFormulasOf</a> <font color="Maroon" title="c2">S</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E9:136"/>
<font color="Maroon" title="c3">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">m</font> <a href="fomodel2.html#K35" title="FOMODEL2:func.35">-NorFormulasOf</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E60">Z0</font></span></a>, <a class="txt" href="#E6:136"><span class="lab"><font color="Green" title="E62">Z1</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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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