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<span class="kw">let </span><font color="Maroon" title="c1">Y</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">  for <font color="Olive" title="b1">R</font> being  <font color="Maroon" title="c1">Y</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> holds   <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">Y</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">Y</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">R</font> be  <font color="Maroon" title="c1">Y</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">Y</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">Y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E2:117"/><span class="lab"><font color="Green" title="E56">B0</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T20" target="_self" title="FOMODEL0:th.20">Th20</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:117_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font><span class="kw"> such that </span><br/><a NAME="E3:117_1"/><span class="lab"><font color="Green" title="E57">C2</font></span>: 
( <font color="Maroon" title="c5">z</font> <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="c6">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">x</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> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E56">B0</font></span></a></span>;<br/><a NAME="E4:117_1"/><span class="lab"><font color="Green" title="E58">C0</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">z</font> <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="c6">x</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1"><span class="lab"><font color="Green" title="E57">C2</font></span></a>, <a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</a></span>;<br/><a NAME="E5:117_1"/>
<font color="Maroon" title="c6">x</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="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1"><span class="lab"><font color="Green" title="E58">C0</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:117_1"/><span class="lab"><font color="Green" title="E59">C1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c3">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>;<br/><a NAME="E8:117_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</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="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:117_1"><span class="lab"><font color="Green" title="E59">C1</font></span></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/><a NAME="E9:117_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="c6">x</font>,<font color="Maroon" title="c6">x</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="c3">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:117_1"><span class="lab"><font color="Green" title="E59">C1</font></span></a>, <a class="ref" href="relat_1.html#D8" title="RELAT_1:def.8">RELAT_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E10:117_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1"><span class="lab"><font color="Green" title="E57">C2</font></span></a>, <a class="ref" href="funct_1.html#T17" title="FUNCT_1:th.17">FUNCT_1:17</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>
<span class="kw">hence </span><a NAME="E4:117"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">Y</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">R</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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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