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<span class="kw">let </span><font color="Maroon" title="c1">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">  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel1.html#V12" title="FOMODEL1:attr.12">-prefix</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">FF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">q1</font>, <font color="Maroon" title="c6">p2</font>, <font color="Maroon" title="c7">q2</font> be   <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">q2</font> implies ( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp; <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:119_1"/><span class="lab"><font color="Green" title="E54">C0</font></span>: 
( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c4">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">q2</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="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp; <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">phi1</font> = <font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c9">phi2</font> = <font color="Maroon" title="c6">p2</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="c1">S</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:119_1"/><span class="lab"><font color="Green" title="E55">C1</font></span>: 
<font color="Maroon" title="c8">phi1</font> is <font color="Maroon" title="c10">m1</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">consider </span><font color="Maroon" title="c11">m2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:119_1"/><span class="lab"><font color="Green" title="E56">C2</font></span>: 
<font color="Maroon" title="c9">phi2</font> is <font color="Maroon" title="c11">m2</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">set </span><font color="Maroon" title="c12">N</font> = <font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">PhiN</font> = <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m2</font></span>)</span>;<br/><a NAME="E7:119_1"/>
( <font color="Maroon" title="c8">phi1</font> is <font color="Maroon" title="c10">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">m2</font></span>)</span> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  &amp; <font color="Maroon" title="c9">phi2</font> is <font color="Maroon" title="c11">m2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">m1</font></span>)</span> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:119_1"><span class="lab"><font color="Green" title="E55">C1</font></span></a>, <a class="txt" href="#E6:119_1"><span class="lab"><font color="Green" title="E56">C2</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">phi11</font> = <font color="Maroon" title="c8">phi1</font>, <font color="Maroon" title="c15">phi22</font> = <font color="Maroon" title="c9">phi2</font> as  <font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m2</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> ;<br/><a NAME="E9:119_1"/>
( <font color="Maroon" title="c14">phi11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m2</font></span>)</span> &amp; <font color="Maroon" title="c15">phi22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#K25" title="FOMODEL2:func.25">-formulasOfMaxDepth</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">m2</font></span>)</span> )
 <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/><span class="kw">hence </span><a NAME="E10:119_1"/>
( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp; <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:119_1"><span class="lab"><font color="Green" title="E54">C0</font></span></a>, <a class="ref" href="fomodel0.html#D20" title="FOMODEL0:def.20">FOMODEL0:def 20</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="E2:119"/><span class="kw">then </span>
 <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> is  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font> <a href="fomodel0.html#V3" title="FOMODEL0:attr.3">-prefix</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#D20" title="FOMODEL0:def.20">FOMODEL0:def 20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:119"/>
 <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel1.html#V12" title="FOMODEL1:attr.12">-prefix</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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