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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="polnot_1.html#NM3" title="POLNOT_1:NM.3">Polish-language</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">E</font> being    <a href="polnot_1.html#M1" title="POLNOT_1:mode.1">Polish-arity-function</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">E</font>) iff  <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="polnot_1.html#M1" title="POLNOT_1:mode.1">Polish-arity-function</a> of <font color="Maroon" title="c1">L</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">  for <font color="Olive" title="b1">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>) iff  <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</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="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>) iff  <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/>





<span class="kw">thus </span><a NAME="E1:173"/>
( <font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>) implies  <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies <font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="173_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:173_1"/><span class="lab"><font color="Green" title="E95">A1</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</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">  <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/>

<a NAME="E2:173_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E3:173_1"/><span class="kw">then </span>
 <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T53" target="_self" onmouseover="rs('polnot_1/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:173_1"/>
 <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:173_1"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</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">assume </span><a NAME="E2:173"/><span class="lab"><font color="Green" title="E95">A10</font></span>: 
 <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span><br/>

<a NAME="E3:173"/><span class="kw">then </span>
<font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="polnot_1.html#K21" title="POLNOT_1:func.21">Polish-WFF-set</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)</span>)</span> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T67" target="_self" onmouseover="rs('polnot_1/T67')" onmouseout="rh()">Th67</a></span>;<br/>
<a NAME="E4:173"/><span class="kw">then </span>
<font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T6" target="_self" onmouseover="rs('polnot_1/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E5:173"/><span class="kw">then </span>
<font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c3">F</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="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:173"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 
;<br/>
<a NAME="E7:173"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:173"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polnot_1.html#K7" title="POLNOT_1:func.7">Polish-atoms</a> (<font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">E</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:173"><span class="lab"><font color="Green" title="E95">A10</font></span></a>, <a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</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>
