<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="finseq_1.html#V4" title="FINSEQ_1:attr.4">FinSequence-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">P</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b2">Q</font> being   <a href="finseq_1.html#V4" title="FINSEQ_1:attr.4">FinSequence-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">Q</font> is <font color="Olive" title="b1">A</font> <a href="polnot_1.html#V2" title="POLNOT_1:attr.2">-closed</a>  holds <br/> <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">P</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Q</font> being   <a href="finseq_1.html#V4" title="FINSEQ_1:attr.4">FinSequence-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">Q</font> is <font color="Maroon" title="c2">A</font> <a href="polnot_1.html#V2" title="POLNOT_1:attr.2">-closed</a>  holds <br/> <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Q</font> be   <a href="finseq_1.html#V4" title="FINSEQ_1:attr.4">FinSequence-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c3">Q</font> is <font color="Maroon" title="c2">A</font> <a href="polnot_1.html#V2" title="POLNOT_1:attr.2">-closed</a>  holds <br/> <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Q</font> is <font color="Maroon" title="c2">A</font> <a href="polnot_1.html#V2" title="POLNOT_1:attr.2">-closed</a>  implies  <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:60"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c3">Q</font> is <font color="Maroon" title="c2">A</font> <a href="polnot_1.html#V2" title="POLNOT_1:attr.2">-closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,$1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>;<br/>
<a NAME="E2:60"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:60_1"/>
 <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D9" target="_self" onmouseover="rs('polnot_1/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:60_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="polnot_1.html#T34" target="_self" onmouseover="rs('polnot_1/T34')" onmouseout="rh()">Th34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:60"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">V</font> =  <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">k</font>);<br/>
<span class="kw">assume </span><a NAME="E1:60_2"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<a NAME="E2:60_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) implies <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font></span><br/>

<a NAME="E2:60_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T23" target="_self" onmouseover="rs('polnot_1/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:60_2_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">and </span><br/><a NAME="E5:60_2_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E6:60_2_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">and </span><br/><a NAME="E7:60_2_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D6" target="_self" onmouseover="rs('polnot_1/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E8:60_2_1"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:60_2_1"/>
<font color="Maroon" title="c9">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Q</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:60_2_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="polnot_1.html#T17" target="_self" onmouseover="rs('polnot_1/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:60_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E4:60_2_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E5:60_2_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E6:60_2_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E8:60_2_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E3:60_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:60"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E43">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:60"/>
 <a href="polnot_1.html#K10" title="POLNOT_1:func.10">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
