<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</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">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b3">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Olive" title="b2">n</font> holds <br/>( <font color="Olive" title="b3">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Olive" title="b1">m</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Olive" title="b3">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c3">n</font> holds <br/>( <font color="Olive" title="b1">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Olive" title="b1">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</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="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c3">n</font> implies ( <font color="Maroon" title="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:104"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c3">n</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="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:104"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/>
<a NAME="E5:104"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E68">A3</font></span></a></span>;<br/>
<a NAME="E6:104"/><span class="kw">then </span>
<font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c3">n</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font> <a href="polnot_1.html#V6" title="POLNOT_1:attr.6">-headed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T51" target="_self" onmouseover="rs('polnot_1/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:104"/>
<font color="Maroon" title="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E67">A2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font>;<br/>
<a NAME="E8:104"/>
<font color="Maroon" title="c4">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="ref" href="polnot_1.html#T10" target="_self" onmouseover="rs('polnot_1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E9:104"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T54" target="_self" onmouseover="rs('polnot_1/T54')" onmouseout="rh()">Th54</a></span>;<br/>
<a NAME="E10:104"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> 1</span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c5">l</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#T10" target="_self" onmouseover="rs('polnot_1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E11:104"/>
<font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c5">l</font></span>)</span> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V6" title="POLNOT_1:attr.6">-headed</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E12:104"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c4">s</font> is <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#V5" title="POLNOT_1:attr.5">-headed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:104"><span class="lab"><font color="Green" title="E70">A6</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>
