<?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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <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="b2">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <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>  <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="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">  <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="c3">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)</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>  <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">$1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>);<br/>
<a NAME="E1:41"/><span class="lab"><font color="Green" title="E28">A1</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="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">U</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>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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>,1);<br/>
<span class="lab"><font color="Green" title="E29">A3</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>,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>,<span class="p1">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)

<br/>.= 
 <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>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>)</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/>
<a NAME="E2:41_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="E3:41_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="#E1:41_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="polnot_1.html#T19" target="_self" onmouseover="rs('polnot_1/T19')" onmouseout="rh()">Th19</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="E2:41"/><span class="lab"><font color="Green" title="E29">A10</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="41_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</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="c4">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="c5">U</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="c4">k</font>);<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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>);<br/>
<a NAME="E2:41_2"/><span class="lab"><font color="Green" title="E31">A13</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">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/>
<a NAME="E3:41_2"/>
 <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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)</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">hence </span><a NAME="E4:41_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">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="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="polnot_1.html#T20" target="_self" onmouseover="rs('polnot_1/T20')" onmouseout="rh()">Th20</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="E3:41"/>
 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="#E1:41"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E29">A10</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:41"/>
 <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="c3">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">n</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"> verum</span><br/>


</div>
