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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">E1</font>, <font color="Olive" title="b2">E2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font>  st  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b1">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b2">E2</font> holds <br/><font color="Olive" title="b1">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">E1</font>, <font color="Maroon" title="c3">E2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">E2</font> implies <font color="Maroon" title="c2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">E2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</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="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E1</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E2</font> ) &amp; ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E2</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E1</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E2</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:8_1_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:8_1_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c4">x</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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:8_1_1"/><span class="lab"><font color="Green" title="E7">A4</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">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:8_1_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a></span>;<br/><a NAME="E7:8_1_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:8_1_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E10:8_1_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/><a NAME="E11:8_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></span>;<br/><a NAME="E12:8_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="c8">b</font>,<font color="Maroon" title="c9">c</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">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/><a NAME="E13:8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c3">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></span>;<br/><a NAME="E14:8_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/><a NAME="E15:8_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="c7">a</font>,<font color="Maroon" title="c9">c</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">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E16:8_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E13:8_1_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:8_1"/><span class="lab"><font color="Green" title="E5">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:8_1"/><span class="lab"><font color="Green" title="E6">A9</font></span>: 
<font color="Maroon" title="c4">x</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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:8_1"/><span class="lab"><font color="Green" title="E7">A10</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">Y</font>
 <span class="kw">and </span><br/><a NAME="E6:8_1"/><span class="lab"><font color="Green" title="E8">A11</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1"><span class="lab"><font color="Green" title="E7">A10</font></span></a>, <a class="txt" href="#E6:8_1"><span class="lab"><font color="Green" title="E8">A11</font></span></a></span>;<br/><a NAME="E8:8_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E5:8_1"><span class="lab"><font color="Green" title="E7">A10</font></span></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:8_1"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">and </span><br/><a NAME="E11:8_1"/><span class="lab"><font color="Green" title="E9">A12</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c2">E1</font>,<font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></span>;<br/><a NAME="E12:8_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1"><span class="lab"><font color="Green" title="E7">A10</font></span></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></span>;<br/><a NAME="E13:8_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="c8">b</font>,<font color="Maroon" title="c9">c</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">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E9">A12</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/><a NAME="E14:8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c2">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></span>;<br/><a NAME="E15:8_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">E2</font>,<font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1"><span class="lab"><font color="Green" title="E5">A8</font></span></a>, <a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E6">A9</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/><a NAME="E16:8_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="c7">a</font>,<font color="Maroon" title="c9">c</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">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E9">A12</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E17:8_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E6">A9</font></span></a>, <a class="txt" href="#E14:8_1"><span class="lab"><font color="Green" title="E10">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:8"/>
<font color="Maroon" title="c2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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