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

<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">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">b1</font>, <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> holds <br/> ( <font color="Olive" title="b1">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Olive" title="b2">b2</font> iff <font color="Olive" title="b1">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Olive" title="b2">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">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="c3">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> iff <font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">B1</font> = <font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B2</font> = <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font>;<br/>
<a NAME="E1:34"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" onmouseover="rs('hilbasis/D1')" onmouseout="rh()">HILBASIS:def 1</a></span>;<br/>
<a NAME="E2:34"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">b1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:34"/>
( <font color="Maroon" title="c3">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> implies <font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</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">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> implies <font color="Maroon" title="c3">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_1"/>
<font color="Maroon" title="c3">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</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">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">o</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font>
 <span class="kw">and </span><br/><a NAME="E4:34_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</font> holds <br/><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D9" onmouseover="rs('pre_poly/D9')" onmouseout="rh()">PRE_POLY:def 9</a></span>;<br/>
<a NAME="E5:34_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c7">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/>
<a NAME="E6:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E7:34_1"/>
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">l</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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="c8">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c8">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font></span><br/>

<a NAME="E2:34_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="ordinal1.html#T10" onmouseover="rs('ordinal1/T10')" onmouseout="rh()">ORDINAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:34_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E19">A4</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="E8:34_1"/>
<font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:34_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="pre_poly.html#D9" onmouseover="rs('pre_poly/D9')" onmouseout="rh()">PRE_POLY:def 9</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="E4:34"/>
<font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</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">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">o</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:34"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font>
 <span class="kw">and </span><br/><a NAME="E7:34"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D9" onmouseover="rs('pre_poly/D9')" onmouseout="rh()">PRE_POLY:def 9</a></span>;<br/>
<a NAME="E8:34"/>
<font color="Maroon" title="c7">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/>
<a NAME="E9:34"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E10:34"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T2" onmouseover="rs('afinsq_1/T2')" onmouseout="rh()">AFINSQ_1:2</a></span>;<br/>
<a NAME="E11:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A10</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T136" onmouseover="rs('zfmisc_1/T136')" onmouseout="rh()">ZFMISC_1:136</a></span>;<br/>
<a NAME="E12:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A11</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E13:34"/>
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</font> holds <br/><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">l</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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="c8">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</font> implies <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_2"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c8">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">o</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">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font></span><br/>

<a NAME="E2:34_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E11:34"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a>, <a class="ref" href="ordinal1.html#T10" onmouseover="rs('ordinal1/T10')" onmouseout="rh()">ORDINAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:34_2"/>
<font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E19">A9</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="E14:34"/>
<font color="Maroon" title="c3">b1</font> <a href="pre_poly.html#R1" title="PRE_POLY:pred.1">&lt;</a> <font color="Maroon" title="c4">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E12:34"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="pre_poly.html#D9" onmouseover="rs('pre_poly/D9')" onmouseout="rh()">PRE_POLY:def 9</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>
