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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  implies <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )</span><br/>

<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1"/>
( <font color="Maroon" title="c1">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> or  not <font color="Maroon" title="c1">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/>
<font color="Maroon" title="c1">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><a NAME="E2:56_1_1"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><b>thus </b><a NAME="E3:56_1_1"/>
<font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:56_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a>,<a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> )</span><br/><b>assume </b><a NAME="E2:56_1_1_1_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font></span><br/><b>hence </b><a NAME="E3:56_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_1_1_1_1"><i><font color="Green" title="E34">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:56_1_1_1_1_2"/><i><font color="Green" title="E34">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E4:56_1_1_1_1_2"/><i><font color="Green" title="E35">A5</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:56_1_1_1_1_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></i>;<br/><b>thus </b><a NAME="E5:56_1_1_1_1_2"/>
<font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_1_1_2_1_1"/>
( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E4:56_1_1_1_1_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E36">A6</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a>,<a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> )</span><br/><b>assume </b><a NAME="E2:56_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font></span><br/><b>hence </b><a NAME="E3:56_1_1_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_1_1_1_2_1_1_1"><i><font color="Green" title="E36">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E36">A7</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a>,<a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> or  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> )</span><br/><b>assume </b><a NAME="E2:56_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E37">A8</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font></span><br/><a NAME="E3:56_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_1_1_1_2_1_1_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:56_1_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1_1_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:56_1_1_1_1_2_1_1_2"><i><font color="Green" title="E37">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/><i><font color="Green" title="E33">A9</font></i>: 
not <font color="Maroon" title="c1">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">X</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st not <font color="Olive" title="b1">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 holds <br/><font color="Olive" title="b1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>;<br/><a NAME="E2:56_1_2"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )</span><br/>

<b>assume </b><a NAME="E1:56_1_2_1"/><i><font color="Green" title="E35">A11</font></i>: 
( not <font color="Maroon" title="c2">X</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c3">R</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E3:56_1_2_1"/><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="neckla_2.html#D4" title="NECKLA_2:def.4">NECKLA_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">cX</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">cX</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E3:56_1_2_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="neckla_2.html#T3" title="NECKLA_2:th.3">NECKLA_2:3</a></i>;<br/>
<b>assume </b><a NAME="E5:56_1_2_1"/><i><font color="Green" title="E37">A13</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/>

<b>assume </b><a NAME="E6:56_1_2_1"/>
not <font color="Maroon" title="c2">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_2_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">cX</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">cX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E5:56_1_2_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="card_2.html#T19" title="CARD_2:th.19">CARD_2:19</a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">cX</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_2_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">cX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="graph_5.html#T5" title="GRAPH_5:th.5">GRAPH_5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:56_1_2"/><i><font color="Green" title="E35">A14</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:56_1_2_2"/><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>let </b><font color="Maroon" title="c3">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )</span><br/>

<b>assume </b><a NAME="E2:56_1_2_2"/><i><font color="Green" title="E37">A16</font></i>: 
( not <font color="Maroon" title="c3">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">k1</font> = <font color="Maroon" title="c2">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">cY</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font>;<br/>
<b>consider </b><font color="Maroon" title="c6">R</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E5:56_1_2_2"/><i><font color="Green" title="E38">A17</font></i>: 
( <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:56_1_2_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="neckla_2.html#D4" title="NECKLA_2:def.4">NECKLA_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">cY</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E5:56_1_2_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="neckla_2.html#T3" title="NECKLA_2:th.3">NECKLA_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">H1</font>, <font color="Maroon" title="c9">H2</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E8:56_1_2_2"/><i><font color="Green" title="E39">A18</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">H2</font> &amp; <font color="Maroon" title="c8">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  &amp; <font color="Maroon" title="c9">H2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_2.html#K4" title="NECKLA_2:func.4">fin_RelStr_sp</a>  )
 <b>and </b><br/><a NAME="E9:56_1_2_2"/><i><font color="Green" title="E40">A19</font></i>: 
( <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c8">H1</font>,<font color="Maroon" title="c9">H2</font> or <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c8">H1</font>,<font color="Maroon" title="c9">H2</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:56_1_2_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="neckla_2.html#T6" title="NECKLA_2:th.6">NECKLA_2:6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">cH1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">cH2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">H2</font>;<br/>
<a NAME="E10:56_1_2_2"/><i><font color="Green" title="E41">A20</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">cY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">H2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E9:56_1_2_2"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a>, <a class="ref" href="neckla_2.html#D3" title="NECKLA_2:def.3">NECKLA_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">R1</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E12:56_1_2_2"/><i><font color="Green" title="E42">A21</font></i>: 
( <font color="Maroon" title="c8">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">R1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="neckla_2.html#D4" title="NECKLA_2:def.4">NECKLA_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">cH1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E12:56_1_2_2"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="neckla_2.html#T3" title="NECKLA_2:th.3">NECKLA_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">R2</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E15:56_1_2_2"/><i><font color="Green" title="E43">A22</font></i>: 
( <font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">R2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">R2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="neckla_2.html#D4" title="NECKLA_2:def.4">NECKLA_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">cH2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">H2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E15:56_1_2_2"><i><font color="Green" title="E43">A22</font></i></a>, <a class="ref" href="neckla_2.html#T3" title="NECKLA_2:th.3">NECKLA_2:3</a></i>;<br/>
<a NAME="E17:56_1_2_2"/><i><font color="Green" title="E44">A23</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="neckla_3.html#E10:56_1_2_2"><i><font color="Green" title="E41">A20</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/>
<b>assume </b><a NAME="E18:56_1_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/>

<a NAME="E19:56_1_2_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">cY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>
<a NAME="E20:56_1_2_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/>
<a NAME="E21:56_1_2_2"/><b>then </b><i><font color="Green" title="E45">A24</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>
<a NAME="E22:56_1_2_2"/>
( not <font color="Maroon" title="c8">H1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c9">H2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a></i>;<br/>
<a NAME="E23:56_1_2_2"/><b>then </b>
( not <font color="Maroon" title="c13">cH1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c15">cH2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E24:56_1_2_2"/><b>then </b>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/>
<a NAME="E25:56_1_2_2"/><b>then </b><i><font color="Green" title="E46">A25</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_2_2_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E17:56_1_2_2"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="neckla_3.html#E21:56_1_2_2"><i><font color="Green" title="E45">A24</font></i></a>, <a class="txt" href="neckla_3.html#E25:56_1_2_2"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><a NAME="E2:56_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E3:56_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><a NAME="E4:56_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><b>hence </b><a NAME="E5:56_1_2_2_1_1"/>
<font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="neckla_3.html#E2:56_1_2_2"><i><font color="Green" title="E37">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_2"/><i><font color="Green" title="E47">A26</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E3:56_1_2_2_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">cY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_2"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/><a NAME="E4:56_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">cY</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:56_1_2_2_1_2"/><i><font color="Green" title="E48">A27</font></i>: 
<font color="Maroon" title="c7">cY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T48" title="CARD_1:th.48">CARD_1:48</a></i>;<br/><b>thus </b><a NAME="E7:56_1_2_2_1_2"/>
<font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:56_1_2_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="neckla_3.html#E6:56_1_2_2_1_2"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_3"/><i><font color="Green" title="E47">A28</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:56_1_2_2_1_3"/><i><font color="Green" title="E48">A29</font></i>: 
<font color="Maroon" title="c13">cH1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:56_1_2_2_1_3"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c15">cH2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_3"><i><font color="Green" title="E47">A28</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E6:56_1_2_2_1_3"/><i><font color="Green" title="E50">A31</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c8">H1</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c13">cH1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:56_1_2_2_1_3"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="neckla_3.html#T5" target="_self" title="NECKLA_3:th.5">Th5</a></i>;<br/><a NAME="E7:56_1_2_2_1_3"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">H2</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c15">cH2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:56_1_2_2_1_3"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="neckla_3.html#T5" target="_self" title="NECKLA_3:th.5">Th5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">H2</font> = <font color="Maroon" title="c9">H2</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">H1</font> = <font color="Maroon" title="c8">H1</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E6:56_1_2_2_1_3"><i><font color="Green" title="E50">A31</font></i></a>, <a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/><a NAME="E10:56_1_2_2_1_3"/>
(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c19">H1</font>,<font color="Maroon" title="c18">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c19">H1</font>,<font color="Maroon" title="c18">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E11:56_1_2_2_1_3"/>
<font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:56_1_2_2"><i><font color="Green" title="E40">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_4"/><i><font color="Green" title="E47">A32</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:56_1_2_2_1_4"/><i><font color="Green" title="E48">A33</font></i>: 
<font color="Maroon" title="c13">cH1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E4:56_1_2_2_1_4"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c8">H1</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c13">cH1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:56_1_2_2_1_4"><i><font color="Green" title="E48">A33</font></i></a>, <a class="ref" href="neckla_3.html#T5" target="_self" title="NECKLA_3:th.5">Th5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">H1</font> = <font color="Maroon" title="c8">H1</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/><a NAME="E6:56_1_2_2_1_4"/>
not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_4"><i><font color="Green" title="E47">A32</font></i></a>, <a class="ref" href="nat_2.html#T30" title="NAT_2:th.30">NAT_2:30</a></i>;<br/><a NAME="E7:56_1_2_2_1_4"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a></i>;<br/><a NAME="E8:56_1_2_2_1_4"/><b>then </b>
( not <font color="Maroon" title="c15">cH2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c15">cH2</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/><a NAME="E9:56_1_2_2_1_4"/><b>then </b>
( not <font color="Maroon" title="c9">H2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c9">H2</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">H2</font> = <font color="Maroon" title="c9">H2</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="neckla_3.html#E1:56_1_2_2_1_4"><i><font color="Green" title="E47">A32</font></i></a></i>;<br/><a NAME="E11:56_1_2_2_1_4"/>
(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c17">H1</font>,<font color="Maroon" title="c18">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c17">H1</font>,<font color="Maroon" title="c18">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E12:56_1_2_2_1_4"/>
<font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:56_1_2_2"><i><font color="Green" title="E40">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_5"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_5"/><i><font color="Green" title="E47">A34</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:56_1_2_2_1_5"/><i><font color="Green" title="E48">A35</font></i>: 
<font color="Maroon" title="c15">cH2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E4:56_1_2_2_1_5"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">H2</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c15">cH2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:56_1_2_2_1_5"><i><font color="Green" title="E48">A35</font></i></a>, <a class="ref" href="neckla_3.html#T5" target="_self" title="NECKLA_3:th.5">Th5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">H2</font> = <font color="Maroon" title="c9">H2</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/><a NAME="E6:56_1_2_2_1_5"/>
not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_5"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="nat_2.html#T30" title="NAT_2:th.30">NAT_2:30</a></i>;<br/><a NAME="E7:56_1_2_2_1_5"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a></i>;<br/><a NAME="E8:56_1_2_2_1_5"/><b>then </b>
( not <font color="Maroon" title="c13">cH1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">cH1</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/><a NAME="E9:56_1_2_2_1_5"/><b>then </b>
( not <font color="Maroon" title="c8">H1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c8">H1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">H1</font> = <font color="Maroon" title="c8">H1</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="neckla_3.html#E1:56_1_2_2_1_5"><i><font color="Green" title="E47">A34</font></i></a></i>;<br/><a NAME="E11:56_1_2_2_1_5"/>
(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c18">H1</font>,<font color="Maroon" title="c17">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c18">H1</font>,<font color="Maroon" title="c17">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E12:56_1_2_2_1_5"/>
<font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:56_1_2_2"><i><font color="Green" title="E40">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_2_1_6"><b>suppose </b></a><a NAME="E1:56_1_2_2_1_6"/><i><font color="Green" title="E47">A36</font></i>: 
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a></span><br/><div class="add"><a NAME="E2:56_1_2_2_1_6"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_2_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_2_2_1_6_1"/>
not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:56_1_2_2_1_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="neckla_3.html#E25:56_1_2_2"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1_2_2_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_6"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E5:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">cH1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><a NAME="E6:56_1_2_2_1_6"/><b>then </b><i><font color="Green" title="E48">A37</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">cH1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E7:56_1_2_2_1_6"/>
not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_6"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="nat_2.html#T30" title="NAT_2:th.30">NAT_2:30</a></i>;<br/><a NAME="E8:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a></i>;<br/><a NAME="E9:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon" title="c13">cH1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c13">cH1</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/><a NAME="E10:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon" title="c8">H1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c8">H1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">H1</font> = <font color="Maroon" title="c8">H1</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="neckla_3.html#E6:56_1_2_2_1_6"><i><font color="Green" title="E48">A37</font></i></a></i>;<br/><a NAME="E12:56_1_2_2_1_6"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_2_1_6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_2_2_1_6_2"/>
not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:56_1_2_2_1_6_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">cH1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="neckla_3.html#E25:56_1_2_2"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1_2_2_1_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_6"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E13:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E15:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">cH2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><a NAME="E16:56_1_2_2_1_6"/><b>then </b><i><font color="Green" title="E49">A38</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">cH2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">k1</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/><a NAME="E17:56_1_2_2_1_6"/>
not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2_1_6"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="nat_2.html#T30" title="NAT_2:th.30">NAT_2:30</a></i>;<br/><a NAME="E18:56_1_2_2_1_6"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">cH2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a></i>;<br/><a NAME="E19:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon" title="c15">cH2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c15">cH2</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/><a NAME="E20:56_1_2_2_1_6"/><b>then </b>
( not <font color="Maroon" title="c9">H2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c9">H2</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">H2</font> = <font color="Maroon" title="c9">H2</font> as  <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E1:56_1_2_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="neckla_3.html#E8:56_1_2_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="neckla_3.html#E16:56_1_2_2_1_6"><i><font color="Green" title="E49">A38</font></i></a></i>;<br/><a NAME="E22:56_1_2_2_1_6"/>
(  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Maroon" title="c16">H1</font>,<font color="Maroon" title="c17">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> &amp;  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> <font color="Maroon" title="c16">H1</font>,<font color="Maroon" title="c17">H2</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> )
 ;<br/><b>hence </b><a NAME="E23:56_1_2_2_1_6"/>
<font color="Maroon" title="c3">Y</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:56_1_2_2"><i><font color="Green" title="E40">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:56_1_2"/><i><font color="Green" title="E36">A39</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="neckla_3.html#E2:56_1_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="neckla_3.html#E3:56_1_2"><i><font color="Green" title="E35">A14</font></i></a>);<br/></i><b>consider </b><font color="Maroon" title="c2">R</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E6:56_1_2"/><i><font color="Green" title="E37">A40</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="neckla_2.html#D4" title="NECKLA_2:def.4">NECKLA_2:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="neckla_3.html#E1:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="neckla_2.html#T4" title="NECKLA_2:th.4">NECKLA_2:4</a></i>;<br/><a NAME="E8:56_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:56_1_2"><i><font color="Green" title="E37">A40</font></i></a>, <a class="ref" href="neckla_2.html#T3" title="NECKLA_2:th.3">NECKLA_2:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:56_1_2"/><i><font color="Green" title="E38">A41</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/><b>thus </b><a NAME="E11:56_1_2"/>
<font color="Maroon" title="c1">X</font> is <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="neckla_3.html#E1:56_1_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="neckla_3.html#E4:56_1_2"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="neckla_3.html#E10:56_1_2"><i><font color="Green" title="E38">A41</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
