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<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</font>);<br/>
<a NAME="E1:106_2_1"/>
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T19" target="_self" onmouseover="rs('polnot_1/T19')" onmouseout="rh()">Th19</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="E2:106_2_1"/>
 not  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</font>) is  <a href="polnot_1.html#V3" title="POLNOT_1:attr.3">antichain-like</a> </span><br/>

<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>  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</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>  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</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>  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</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>  <a href="polnot_1.html#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</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="E3:106_2_1"/><span class="lab"><font color="Green" title="E67">A11</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#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</font>)
 <span class="kw">and </span><br/><a NAME="E4:106_2_1"/><span class="lab"><font color="Green" title="E68">A12</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#K6" title="POLNOT_1:func.6">Polish-expression-layer</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">U</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">t1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c8">u1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a> <span class="kw"> such that </span><br/><a NAME="E6:106_2_1"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">u1</font>
 <span class="kw">and </span><br/><a NAME="E7:106_2_1"/><span class="lab"><font color="Green" title="E70">A14</font></span>: 
<font color="Maroon" title="c8">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polnot_1.html#K14" title="POLNOT_1:func.14">.</a> <font color="Maroon" title="c7">t1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_2_1"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="ref" href="polnot_1.html#D19" target="_self" onmouseover="rs('polnot_1/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">t2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c10">u2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> <a href="polnot_1.html#K5" title="POLNOT_1:func.5">*</a> <span class="kw"> such that </span><br/><a NAME="E9:106_2_1"/><span class="lab"><font color="Green" title="E71">A15</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t2</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">u2</font>
 <span class="kw">and </span><br/><a NAME="E10:106_2_1"/><span class="lab"><font color="Green" title="E72">A16</font></span>: 
<font color="Maroon" title="c10">u2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="polnot_1.html#K14" title="POLNOT_1:func.14">.</a> <font color="Maroon" title="c9">t2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106_2_1"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="polnot_1.html#D19" target="_self" onmouseover="rs('polnot_1/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<a NAME="E11:106_2_1"/>
<font color="Maroon" title="c7">t1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t2</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a>, <a class="txt" href="#E6:106_2_1"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="txt" href="#E9:106_2_1"><span class="lab"><font color="Green" title="E71">A15</font></span></a></span>;<br/>
<a NAME="E12:106_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t2</font> &amp; <font color="Maroon" title="c8">u1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T43" target="_self" onmouseover="rs('polnot_1/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:106_2_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="#E7:106_2_1"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="txt" href="#E10:106_2_1"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="polnot_1.html#D16" target="_self" onmouseover="rs('polnot_1/D16')" onmouseout="rh()">Def16</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/>


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