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<span class="kw">set </span><font color="Maroon" title="c5">IU</font> = <font color="Maroon" title="c2">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">III</font> = <font color="Maroon" title="c3">I</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">II</font> = <font color="Maroon" title="c3">I</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:77_1_1"/><span class="lab"><font color="Green" title="E37">B1</font></span>: 
<font color="Maroon" title="c4">w</font> is <font color="Maroon" title="c9">n</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" target="_self" title="FOMODEL2:def.25">DefWff2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">nn</font> = <font color="Maroon" title="c9">n</font> as    <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">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">fn</font> = (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">ww</font> = <font color="Maroon" title="c4">w</font> as  <font color="Maroon" title="c9">n</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-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="#E2:77_1_1"><span class="lab"><font color="Green" title="E37">B1</font></span></a></span>;<br/>
<a NAME="E5:77_1_1"/><span class="lab"><font color="Green" title="E38">B4</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c10">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c10">nn</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D23" target="_self" title="FOMODEL2:def.23">DefFormula1</a></span>;<br/>
<a NAME="E6:77_1_1"/>
<font color="Maroon" title="c12">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_1"><span class="lab"><font color="Green" title="E38">B4</font></span></a>, <a class="ref" href="fomodel2.html#D24" target="_self" title="FOMODEL2:def.24">DefWff1</a></span>;<br/>
<a NAME="E7:77_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">ww</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">IT</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ww</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">IT</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">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c4">w</font> is <font color="Olive" title="b1">m</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  holds <br/><font color="Maroon" title="c13">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">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"> ( <font color="Maroon" title="c4">w</font> is <font color="Maroon" title="c14">m</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  implies <font color="Maroon" title="c13">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">fm</font> = (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">f</font> = (<font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">g</font> = (<font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">m</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">mm</font> = <font color="Maroon" title="c14">m</font>, <font color="Maroon" title="c19">nn</font> = <font color="Maroon" title="c9">n</font> as    <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">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E10:77_1_1"/><span class="lab"><font color="Green" title="E39">B3</font></span>: 
( <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c18">mm</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> &amp; <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <font color="Maroon" title="c19">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c19">nn</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D23" target="_self" title="FOMODEL2:def.23">DefFormula1</a></span>;<br/>
<a NAME="E11:77_1_1"/><span class="lab"><font color="Green" title="E40">B2</font></span>: 
( (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> (<font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a>  &amp; (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> (<font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">m</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a>  &amp; (<font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> (<font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">m</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel2.html#E36"><span class="lab"><font color="Green" title="E29">Lm29</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:77_1_1"/>
<font color="Maroon" title="c4">w</font> is <font color="Maroon" title="c14">m</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</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="c13">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font></span><br/>

<a NAME="E13:77_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77_1_1"><span class="lab"><font color="Green" title="E39">B3</font></span></a>, <a class="ref" href="fomodel2.html#D24" target="_self" title="FOMODEL2:def.24">DefWff1</a></span>;<br/>
<a NAME="E14:77_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">B6</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_1"><span class="lab"><font color="Green" title="E40">B2</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>;<br/>
<a NAME="E15:77_1_1"/>
<font color="Maroon" title="c12">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c9">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_1"><span class="lab"><font color="Green" title="E38">B4</font></span></a>, <a class="ref" href="fomodel2.html#D24" target="_self" title="FOMODEL2:def.24">DefWff1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:77_1_1"/>
<font color="Maroon" title="c13">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c14">m</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:77_1_1"><span class="lab"><font color="Green" title="E41">B6</font></span></a>, <a class="txt" href="#E11:77_1_1"><span class="lab"><font color="Green" title="E40">B2</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</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>
