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<span class="kw">set </span><font color="Maroon" title="c3">Y</font> =  <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>);<br/>
<a NAME="E1:111_1_1"/>
 <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T26" target="_self" onmouseover="rs('polnot_1/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Y</font> =  <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a></span>)</span> ;<br/>
<a NAME="E3:111_1_1"/>
<font color="Maroon" title="c4">Y</font> is  <a href="polnot_1.html#V3" title="POLNOT_1:attr.3">antichain-like</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="polnot_1.html#D16" target="_self" onmouseover="rs('polnot_1/D16')" onmouseout="rh()">POLNOT_1:def 16</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#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> holds <br/><font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">q</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:111_1_1_1"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:111_1_1_1"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Y</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" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:111_1_1_1"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</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="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="polnot_1.html#T28" target="_self" onmouseover="rs('polnot_1/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:111_1_1_1"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">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="#E2:111_1_1_1"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="ref" href="polnot_1.html#T28" target="_self" onmouseover="rs('polnot_1/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">B</font> =  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>);<br/>
<a NAME="E7:111_1_1_1"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
 <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</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#T25" target="_self" onmouseover="rs('polnot_1/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<a NAME="E8:111_1_1_1"/><span class="lab"><font color="Green" title="E72">A13</font></span>: 
 <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</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#T25" target="_self" onmouseover="rs('polnot_1/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<a NAME="E9:111_1_1_1"/>
 <a href="polnot_1.html#K19" title="POLNOT_1:func.19">Polish-expression-hierarchy</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</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> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>) is  <a href="polnot_1.html#V3" title="POLNOT_1:attr.3">antichain-like</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E10:111_1_1_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_1_1_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="txt" href="#E6:111_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E7:111_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E8:111_1_1_1"><span class="lab"><font color="Green" title="E72">A13</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="E4:111_1_1"/>
 <a href="polnot_1.html#K11" title="POLNOT_1:func.11">Polish-expression-set</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>) is   <a href="polnot_1.html#NM4" title="POLNOT_1:NM.4">Polish-language</a> of <font color="Maroon" title="c1">T</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/>


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