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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">PA1</font>, <font color="Olive" title="b2">PA2</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">PA1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">PA2</font> holds <br/>for <font color="Olive" title="b3">p2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">PA2</font>  ex <font color="Olive" title="b4">p1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">PA1</font> st <font color="Olive" title="b4">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">p2</font></span><br/><b>let </b><font color="Maroon" title="c2">PA1</font>, <font color="Maroon" title="c3">PA2</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">PA1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">PA2</font> implies for <font color="Olive" title="b1">p2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PA2</font>  ex <font color="Olive" title="b2">p1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA1</font> st <font color="Olive" title="b2">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p2</font> )</span><br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">PA1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">PA2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PA2</font>  ex <font color="Olive" title="b2">p1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA1</font> st <font color="Olive" title="b2">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p2</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">E1</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:9"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">PA1</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="c4">E1</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T43" title="EQREL_1:th.43">EQREL_1:43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">E2</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E5:9"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">PA2</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="c5">E2</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T43" title="EQREL_1:th.43">EQREL_1:43</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">p2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PA2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA1</font> st <font color="Olive" title="b1">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">p2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">p2'</font> = <font color="Maroon" title="c6">p2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:9"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">p2'</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="c5">E2</font>,<font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="txt" href="#E5:9"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">E1a</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">E1</font>,<font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA1</font> <b>by </b><i><a class="txt" href="#E3:9"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E8:9"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">E1a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">E1a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">p2</font></span><br/>

<b>consider </b><font color="Maroon" title="c10">p22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:9"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c10">p22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA2</font> &amp; <font color="Maroon" title="c9">E1a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">p22</font> )
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">p22'</font> = <font color="Maroon" title="c10">p22</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="#E11:9"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:9"/><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">p22'</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="c5">E2</font>,<font color="Maroon" title="c12">b</font> )
 <b>by </b><i><a class="txt" href="#E5:9"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E11:9"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E15:9"/>
<font color="Maroon" title="c8">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="c4">E1</font>,<font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="#E8:9"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<b>hence </b><a NAME="E16:9"/>
<font color="Maroon" title="c9">E1a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">p2</font>
 <b>by </b><i><a class="txt" href="#E8:9"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E11:9"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E14:9"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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