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<span class="kw">set </span><font color="Maroon" title="c3">X</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">set </span><font color="Maroon" title="c4">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</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> ;<br/>
<a NAME="E1:43_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</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>  or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</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> 
 ; <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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:43_1_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c6">Z</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="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="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:43_1_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c6">Z</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="c7">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:43_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E6:43_1_1_1"><span class="lab"><font color="Green" title="E31">A3</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="E2:43_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</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>  is   <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">P</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a></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/>


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