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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <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">A</font> being    <a href="polnot_1.html#M1" title="POLNOT_1:mode.1">Polish-arity-function</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">f</font> being   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b4">K1</font>, <font color="Olive" title="b5">K2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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">T</font>,<font color="Olive" title="b1">A</font>)</span>)</span>,<font color="Olive" title="b2">D</font><br/>  for <font color="Olive" title="b6">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">A</font>  st <font color="Olive" title="b4">K1</font> is <font color="Olive" title="b3">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b5">K2</font> is <font color="Olive" title="b3">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b6">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">T</font>,<font color="Olive" title="b1">A</font>) holds <br/><font color="Olive" title="b4">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <a href="polnot_1.html#M1" title="POLNOT_1:mode.1">Polish-arity-function</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">  for <font color="Olive" title="b1">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">f</font> being   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b3">K1</font>, <font color="Olive" title="b4">K2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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">T</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>,<font color="Olive" title="b1">D</font><br/>  for <font color="Olive" title="b5">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b3">K1</font> is <font color="Olive" title="b2">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b4">K2</font> is <font color="Olive" title="b2">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b5">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">T</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Olive" title="b3">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">f</font> being   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b2">K1</font>, <font color="Olive" title="b3">K2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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">T</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b4">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">K1</font> is <font color="Olive" title="b1">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b3">K2</font> is <font color="Olive" title="b1">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b4">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">T</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Olive" title="b2">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="polnot_1.html#NM7" title="POLNOT_1:NM.7">Polish-recursion-function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">D</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">K1</font>, <font color="Olive" title="b2">K2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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">T</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</font><br/>  for <font color="Olive" title="b3">F</font> being   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">K1</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b2">K2</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Olive" title="b3">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">T</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Olive" title="b1">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">K1</font>, <font color="Maroon" title="c6">K2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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">T</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>,<font color="Maroon" title="c3">D</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">T</font>,<font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c5">K1</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Maroon" title="c6">K2</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <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">T</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c5">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</font> be   <a href="polnot_1.html#NM6" title="POLNOT_1:NM.6">Polish-WFF</a> of <font color="Maroon" title="c1">T</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"> ( <font color="Maroon" title="c5">K1</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Maroon" title="c6">K2</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a>  &amp; <font color="Maroon" title="c7">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">T</font>,<font color="Maroon" title="c2">A</font>) implies <font color="Maroon" title="c5">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:152"/><span class="lab"><font color="Green" title="E83">A1</font></span>: 
<font color="Maroon" title="c5">K1</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a> 
 <span class="kw">and </span><br/><a NAME="E2:152"/><span class="lab"><font color="Green" title="E84">A2</font></span>: 
<font color="Maroon" title="c6">K2</font> is <font color="Maroon" title="c4">f</font> <a href="polnot_1.html#V8" title="POLNOT_1:attr.8">-recursive</a> 
 <span class="kw">and </span><br/><a NAME="E3:152"/><span class="lab"><font color="Green" title="E85">A3</font></span>: 
<font color="Maroon" title="c7">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">T</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"> <font color="Maroon" title="c5">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font></span><br/>

<a NAME="E4:152"/>
<font color="Maroon" title="c7">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E85">A3</font></span></a>, <a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E5:152"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A5</font></span>: 
 <a href="polnot_1.html#K29" title="POLNOT_1:func.29">Polish-WFF-head</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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/>
<a NAME="E6:152"/><span class="kw">then </span>
 <a href="polnot_1.html#K31" title="POLNOT_1:func.31">Polish-arity</a> <font color="Maroon" title="c7">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="#E3:152"><span class="lab"><font color="Green" title="E85">A3</font></span></a>, <a class="ref" href="polnot_1.html#D7" target="_self" onmouseover="rs('polnot_1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E7:152"/><span class="kw">then </span>
 <a href="polnot_1.html#K35" title="POLNOT_1:func.35">Polish-WFF-args</a> <font color="Maroon" title="c7">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="polnot_1.html#T61" target="_self" onmouseover="rs('polnot_1/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E8:152"/><span class="kw">then </span>
( <font color="Maroon" title="c5">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">K1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E84">A2</font></span></a>, <a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E86">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:152"/>
<font color="Maroon" title="c5">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">F</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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