<?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">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">A</font>) holds <br/> ex <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">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="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">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">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#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>) holds <br/> ex <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">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="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="c3">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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</font>) implies  ex <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">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="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">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">A</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">  ex <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">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="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">set </span><font color="Maroon" title="c4">Y</font> = <span class="p1"> { <span class="default"> <span class="p2">(<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="Olive" title="b1">n</font>)</span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<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="Olive" title="b1">n</font>)</span>)</span> where <font color="Olive" title="b1">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:46"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<font color="Maroon" title="c6">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E34">A3</font></span></a></span>;<br/>
<a NAME="E7:46"/>
 <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="c6">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="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T24" target="_self" onmouseover="rs('polnot_1/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:46"/>
 ex <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">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="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E35">A4</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>
