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<span class="kw">consider </span><font color="Maroon" title="c2">R9</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><span class="kw"> such that </span><br/><a NAME="E2:7_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c2">R9</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T17" title="WELLORD2:th.17">WELLORD2:17</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c2">R9</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E3:7_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c2">R9</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">A</font> is  <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord2.html#T16" title="WELLORD2:th.16">WELLORD2:16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">R</font> = <font color="Maroon" title="c2">R9</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">A</font> as   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">R</font> )</span><br/><span class="kw">assume </span><a NAME="E1:7_1_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">R</font></span><br/><a NAME="E2:7_1_1_1"/>
<font color="Maroon" title="c2">R9</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></span>;<br/><a NAME="E3:7_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">R9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></span>;<br/><a NAME="E4:7_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><a NAME="E5:7_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:7_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E7:7_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">R</font> = <font color="Maroon" title="c4">R</font> as   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="wellord1.html#D4" title="WELLORD1:def.4">WELLORD1:def 4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> </span><br/>

<span class="kw">thus </span><a NAME="E9:7_1_1"/>
<font color="Maroon" title="c5">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="wellord1.html#D4" title="WELLORD1:def.4">WELLORD1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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