<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">H</font> being   <a href="abian.html#V2" title="ABIAN:attr.2">covering</a>   <a href="taxonom2.html#V5" title="TAXONOM2:attr.5">T_3</a>   <a href="taxonom2.html#M1" title="TAXONOM2:mode.1">Hierarchy</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">H</font> is  <a href="taxonom2.html#V6" title="TAXONOM2:attr.6">lower-bounded</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; (  for <font color="Olive" title="b2">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">P2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C1</font> &amp; <font color="Olive" title="b4">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C1</font> &amp;  not <font color="Olive" title="b3">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b4">P2</font> holds <br/><font color="Olive" title="b4">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">P1</font> ) holds <br/> ex <font color="Olive" title="b3">PX</font>, <font color="Olive" title="b4">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">PX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C1</font> &amp; <font color="Olive" title="b4">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C1</font> &amp; (  for <font color="Olive" title="b5">PZ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C1</font> holds <br/>( <font color="Olive" title="b5">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b4">PY</font> &amp; <font color="Olive" title="b3">PX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b5">PZ</font> ) ) ) ) holds <br/> ex <font color="Olive" title="b2">C</font> being    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font> st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="abian.html#V2" title="ABIAN:attr.2">covering</a>   <a href="taxonom2.html#V5" title="TAXONOM2:attr.5">T_3</a>   <a href="taxonom2.html#M1" title="TAXONOM2:mode.1">Hierarchy</a> of <font color="Maroon" title="c1">X</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="c2">H</font> is  <a href="taxonom2.html#V6" title="TAXONOM2:attr.6">lower-bounded</a>  &amp;  not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> &amp; (  for <font color="Olive" title="b1">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">P2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Olive" title="b3">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp;  not <font color="Olive" title="b2">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">P2</font> holds <br/><font color="Olive" title="b3">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">P1</font> ) holds <br/> ex <font color="Olive" title="b2">PX</font>, <font color="Olive" title="b3">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">PX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Olive" title="b3">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; (  for <font color="Olive" title="b4">PZ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> holds <br/>( <font color="Olive" title="b4">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">PY</font> &amp; <font color="Olive" title="b2">PX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b4">PZ</font> ) ) ) ) implies  ex <font color="Olive" title="b1">C</font> being    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font> st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">H</font> is  <a href="taxonom2.html#V6" title="TAXONOM2:attr.6">lower-bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">and </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 for <font color="Olive" title="b1">C1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">C1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> &amp; (  for <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">P2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Olive" title="b3">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp;  not <font color="Olive" title="b2">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">P2</font> holds <br/><font color="Olive" title="b3">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">P1</font> ) holds <br/> ex <font color="Olive" title="b2">PX</font>, <font color="Olive" title="b3">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">PX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; <font color="Olive" title="b3">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> &amp; (  for <font color="Olive" title="b4">PZ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> holds <br/>( <font color="Olive" title="b4">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b3">PY</font> &amp; <font color="Olive" title="b2">PX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b4">PZ</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font> st  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font></span><br/>

<a NAME="E4:37"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T4" title="ABIAN:th.4">ABIAN:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:37"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">h</font> = <font color="Maroon" title="c3">h</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">PX</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E9:37"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">PX</font>
 <span class="kw">and </span><br/><a NAME="E10:37"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c5">PX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="taxonom2.html#T17" target="_self" title="TAXONOM2:th.17">Th17</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">L</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E11:37"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">PX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_1"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1: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/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E30">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">P1</font>, <font color="Maroon" title="c8">P2</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="c7">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c8">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c7">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c8">P2</font> implies <font color="Maroon" title="c8">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c7">P1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_2"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c7">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:37_2"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c8">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c7">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c8">P2</font> or <font color="Maroon" title="c8">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c7">P1</font> )</span><br/><a NAME="E3:37_2"/>
<font color="Maroon" title="c7">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">PX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:37_2"/>
( <font color="Maroon" title="c7">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c8">P2</font> or <font color="Maroon" title="c8">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c7">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E13:37"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c2">H</font> is  <a href="taxonom2.html#V3" title="TAXONOM2:attr.3">hierarchic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="taxonom2.html#D4" target="_self" title="TAXONOM2:def.4">Def4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( (  for <font color="Olive" title="b1">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/><font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font> ) &amp; (  for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b2">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp;  not <font color="Olive" title="b1">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">P2</font> holds <br/><font color="Olive" title="b2">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">P1</font> ) );<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">RL</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:37"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
 for <font color="Olive" title="b1">L</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font> iff ( <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">L</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E16:37"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E17:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">PX</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E11:37"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="c8">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">RL</font> &amp; <font color="Maroon" title="c8">Z</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  implies  ex <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font> &amp; (  for <font color="Olive" title="b2">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_3"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c8">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">RL</font>
 <span class="kw">and </span><br/><a NAME="E2:37_3"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<font color="Maroon" title="c8">Z</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font> &amp; (  for <font color="Olive" title="b2">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c9">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font>;<br/><div><span class="lab"><font color="Green" title="E36">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">X1</font>, <font color="Maroon" title="c11">X2</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="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c11">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp;  not <font color="Maroon" title="c10">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X2</font> implies <font color="Maroon" title="c11">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">X1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_3_1"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c10">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">and </span><br/><a NAME="E2:37_3_1"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<font color="Maroon" title="c11">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</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="c10">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X2</font> or <font color="Maroon" title="c11">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">X1</font> )</span><br/><a NAME="E3:37_3_1"/>
<font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c11">X2</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_3"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E1:37_3_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E2:37_3_1"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:37_3_1"/>
( <font color="Maroon" title="c10">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X2</font> or <font color="Maroon" title="c11">X2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">X1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</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><div><span class="lab"><font color="Green" title="E37">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">P1</font>, <font color="Maroon" title="c11">P2</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="c10">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c11">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font> &amp;  not <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_3_2"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c10">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">and </span><br/><a NAME="E2:37_3_2"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c11">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c12">L1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:37_3_2"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c10">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">L1</font>
 <span class="kw">and </span><br/><a NAME="E5:37_3_2"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<font color="Maroon" title="c12">L1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_3_2"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">L2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:37_3_2"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<font color="Maroon" title="c11">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">L2</font>
 <span class="kw">and </span><br/><a NAME="E8:37_3_2"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<font color="Maroon" title="c13">L2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_3_2"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_3_2_1"/>
( <font color="Maroon" title="c12">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L2</font> or <font color="Maroon" title="c13">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">L1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_3"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E5:37_3_2"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E8:37_3_2"><span class="lab"><font color="Green" title="E43">A24</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_3_2_1_1"/>
<font color="Maroon" title="c12">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_3_2_1_1"/>
( <font color="Maroon" title="c10">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P2</font> or <font color="Maroon" title="c11">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c10">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:37_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E4:37_3_2"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E7:37_3_2"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E8:37_3_2"><span class="lab"><font color="Green" title="E43">A24</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_3_2_1_2"/>
<font color="Maroon" title="c13">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">L1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_3_2_1_2"/>
( <font color="Maroon" title="c10">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P2</font> or <font color="Maroon" title="c11">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c10">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:37_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E4:37_3_2"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E5:37_3_2"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E7:37_3_2"><span class="lab"><font color="Green" title="E42">A23</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">take </span><font color="Maroon" title="c10">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">Z</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="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font> &amp; (  for <font color="Olive" title="b1">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> ) )</span><br/><div><span class="lab"><font color="Green" title="E38">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">P</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="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> implies <font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:37_3_3"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</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="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><a NAME="E2:37_3_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">L3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L3</font> &amp; <font color="Olive" title="b1">L3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_3_3"/>
<font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:37_3"><span class="lab"><font color="Green" title="E34">A13</font></span></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="E6:37_3"/>
<font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> or <font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_3_4"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</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="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">L4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_3_4"/><span class="lab"><font color="Green" title="E39">A26</font></span>: 
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">L4</font>
 <span class="kw">and </span><br/><a NAME="E4:37_3_4"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c12">L4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:37_3_4"/>
<font color="Maroon" title="c12">L4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:37_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E4:37_3_4"><span class="lab"><font color="Green" title="E40">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:37_3_4"/>
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_3_4"><span class="lab"><font color="Green" title="E39">A26</font></span></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="E7:37_3"/>
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E5:37_3"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E4:37_3"><span class="lab"><font color="Green" title="E37">A18</font></span></a></span>; <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">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font></span><br/><span class="kw">thus </span><a NAME="E8:37_3"/>
 for <font color="Olive" title="b1">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" title="ZFMISC_1:th.74">ZFMISC_1:74</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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E20:37"/><span class="lab"><font color="Green" title="E34">A28</font></span>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font>
 <span class="kw">and </span><br/><a NAME="E21:37"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
 for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">C</font> holds <br/> not <font color="Maroon" title="c8">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="orders_1.html#T65" title="ORDERS_1:th.65">ORDERS_1:65</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></a></span>;<br/>
<a NAME="E23:37"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
<font color="Maroon" title="c9">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">P1</font>, <font color="Maroon" title="c11">P2</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or  not <font color="Maroon" title="c11">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c10">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P2</font> or <font color="Maroon" title="c11">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c10">P1</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_4"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
<font color="Maroon" title="c10">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">and </span><br/><a NAME="E2:37_4"/><span class="lab"><font color="Green" title="E38">A32</font></span>: 
<font color="Maroon" title="c11">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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="c10">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P2</font> or <font color="Maroon" title="c11">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c10">P1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:37_4"/>
( <font color="Maroon" title="c10">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c11">P2</font> or <font color="Maroon" title="c11">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c10">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E1:37_4"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="txt" href="#E2:37_4"><span class="lab"><font color="Green" title="E38">A32</font></span></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="E24:37"/><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E21:37"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<a NAME="E25:37"/><span class="lab"><font color="Green" title="E38">A34</font></span>: 
<font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> or <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5"/><span class="lab"><font color="Green" title="E39">A35</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</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="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1"/>
(  not <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1"/><span class="lab"><font color="Green" title="E40">A36</font></span>: 
 not <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</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="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">hx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">hx</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hx</font> );<br/><span class="kw">consider </span><font color="Maroon" title="c11">Ca</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_5_1_1"/><span class="lab"><font color="Green" title="E41">A37</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">p</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span><a NAME="E4:37_5_1_1"/><span class="lab"><font color="Green" title="E42">A38</font></span>: 
<font color="Maroon" title="c11">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> or <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</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="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font></span><br/>

<span class="kw">hence </span><a NAME="E2:37_5_1_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1"><span class="lab"><font color="Green" title="E41">A37</font></span></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">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">hx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b1">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hx</font> );<br/><span class="kw">consider </span><font color="Maroon" title="c12">Cb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:37_5_1_1"/><span class="lab"><font color="Green" title="E43">A39</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span><span class="kw">consider </span><font color="Maroon" title="c13">PS</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E8:37_5_1_1"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">PS</font>
 <span class="kw">and </span><br/><a NAME="E9:37_5_1_1"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c13">PS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="taxonom2.html#T17" target="_self" title="TAXONOM2:th.17">Th17</a></span>;<br/><a NAME="E10:37_5_1_1"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_5_1_1"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="ref" href="eqrel_1.html#D4" title="EQREL_1:def.4">EQREL_1:def 4</a></span>;<br/><div><span class="lab"><font color="Green" title="E47">A43</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c14">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:37_5_1_1_2"/><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_5_1_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c15">p</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="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> implies  ex <font color="Olive" title="b1">hv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp;  not <font color="Olive" title="b1">hv</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font> ) )</span><br/><span class="kw">assume </span><a NAME="E3:37_5_1_1_2"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">hv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> &amp;  not <font color="Olive" title="b1">hv</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font> )</span><br/><a NAME="E4:37_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">p</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E5:37_5_1_1_2"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D4" title="EQREL_1:def.4">EQREL_1:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:37_5_1_1_2"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">v</font>
 <span class="kw">and </span><br/><a NAME="E8:37_5_1_1_2"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E2:37_5_1_1_2"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">assume </span><a NAME="E9:37_5_1_1_2"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
 for <font color="Olive" title="b1">hv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">p</font> holds <br/><font color="Olive" title="b1">hv</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E10:37_5_1_1_2"/>
 not <font color="Maroon" title="c16">v</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_2"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E7:37_5_1_1_2"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E11:37_5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_5_1_1_2"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="txt" href="#E8:37_5_1_1_2"><span class="lab"><font color="Green" title="E50">A46</font></span></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="E12:37_5_1_1"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_3"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c15">hv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_5_1_1_3"/><span class="lab"><font color="Green" title="E50">A50</font></span>: 
<font color="Maroon" title="c15">hv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">and </span><br/><a NAME="E4:37_5_1_1_3"/><span class="lab"><font color="Green" title="E51">A51</font></span>: 
 not <font color="Maroon" title="c15">hv</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_5_1_1"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E1:37_5_1_1_3"><span class="lab"><font color="Green" title="E49">A49</font></span></a></span>;<br/>
<a NAME="E5:37_5_1_1_3"/><span class="lab"><font color="Green" title="E52">A52</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">hv</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="txt" href="#E1:37_5_1_1_3"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E3:37_5_1_1_3"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E6:37_5_1_1_3"/><span class="lab"><font color="Green" title="E53">A53</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E1:37_5_1_1_3"><span class="lab"><font color="Green" title="E49">A49</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_3_1"/>
( <font color="Maroon" title="c15">hv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> or <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">hv</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E3:37_5_1_1_3"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="txt" href="#E4:37_5_1_1_3"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="txt" href="#E6:37_5_1_1_3"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_3_1_1"/>
<font color="Maroon" title="c15">hv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</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="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font></span><br/><div class="add"><a NAME="E2:37_5_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_5_1_1"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="txt" href="#E1:37_5_1_1_3"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E3:37_5_1_1_3"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="txt" href="#E5:37_5_1_1_3"><span class="lab"><font color="Green" title="E52">A52</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_3_1_1"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_3_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">hv</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="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font></span><br/><div class="add"><a NAME="E2:37_5_1_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="txt" href="#E1:37_5_1_1_3"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E3:37_5_1_1_3"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="txt" href="#E5:37_5_1_1_3"><span class="lab"><font color="Green" title="E52">A52</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_3_1_2"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E13:37_5_1_1"/>
<font color="Maroon" title="c12">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font> or <font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_4"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</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="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font></span><br/>

<span class="kw">hence </span><a NAME="E2:37_5_1_1_4"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_5_1_1"><span class="lab"><font color="Green" title="E43">A39</font></span></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="E14:37_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_5_1_1"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E15:37_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A54</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Ca</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_5_1_1"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E16:37_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A55</font></span>: 
<font color="Maroon" title="c11">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><div><span class="lab"><font color="Green" title="E51">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">P1</font>, <font color="Maroon" title="c15">P2</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="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> &amp;  not <font color="Maroon" title="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> implies <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_5_1_1_5"/><span class="lab"><font color="Green" title="E52">A57</font></span>: 
<font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font>
 <span class="kw">and </span><br/><a NAME="E2:37_5_1_1_5"/><span class="lab"><font color="Green" title="E53">A58</font></span>: 
<font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</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="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> or <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )</span><br/><a NAME="E3:37_5_1_1_5"/>
<font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E2:37_5_1_1_5"><span class="lab"><font color="Green" title="E53">A58</font></span></a></span>;<br/><a NAME="E4:37_5_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A59</font></span>: 
<font color="Maroon" title="c15">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E5:37_5_1_1_5"/>
<font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E1:37_5_1_1_5"><span class="lab"><font color="Green" title="E52">A57</font></span></a></span>;<br/><a NAME="E6:37_5_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c14">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:37_5_1_1_5"/>
( <font color="Maroon" title="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> or <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E1:37_5_1_1_5"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E2:37_5_1_1_5"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="txt" href="#E4:37_5_1_1_5"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</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="E18:37_5_1_1"/><span class="lab"><font color="Green" title="E52">A60</font></span>: 
<font color="Maroon" title="c12">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_5_1_1"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><div><span class="lab"><font color="Green" title="E53">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">P1</font>, <font color="Maroon" title="c15">P2</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="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font> &amp;  not <font color="Maroon" title="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> implies <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_5_1_1_6"/><span class="lab"><font color="Green" title="E54">A62</font></span>: 
<font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">and </span><br/><a NAME="E2:37_5_1_1_6"/><span class="lab"><font color="Green" title="E55">A63</font></span>: 
<font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</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="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> or <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )</span><br/><a NAME="E3:37_5_1_1_6"/>
<font color="Maroon" title="c15">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_5_1_1"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="txt" href="#E2:37_5_1_1_6"><span class="lab"><font color="Green" title="E55">A63</font></span></a></span>;<br/><a NAME="E4:37_5_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A64</font></span>: 
<font color="Maroon" title="c15">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E5:37_5_1_1_6"/>
<font color="Maroon" title="c14">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_5_1_1"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="txt" href="#E1:37_5_1_1_6"><span class="lab"><font color="Green" title="E54">A62</font></span></a></span>;<br/><a NAME="E6:37_5_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c14">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:37_5_1_1_6"/>
( <font color="Maroon" title="c14">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">P2</font> or <font color="Maroon" title="c15">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E18:37_5_1_1"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="txt" href="#E1:37_5_1_1_6"><span class="lab"><font color="Green" title="E54">A62</font></span></a>, <a class="txt" href="#E2:37_5_1_1_6"><span class="lab"><font color="Green" title="E55">A63</font></span></a>, <a class="txt" href="#E4:37_5_1_1_6"><span class="lab"><font color="Green" title="E56">A64</font></span></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1"/>
( <font color="Maroon" title="c12">Cb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">Cb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E54">A65</font></span>: 
<font color="Maroon" title="c12">Cb</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">h</font>;<br/><a NAME="E2:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E55">A66</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_1_1"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font></span><br/>

<span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_1_1"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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">consider </span><font color="Maroon" title="c14">PX</font>, <font color="Maroon" title="c15">Pmax</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:37_5_1_1_7_1_1"/>
<font color="Maroon" title="c14">PX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">and </span><br/><a NAME="E5:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E56">A67</font></span>: 
<font color="Maroon" title="c15">Pmax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font>
 <span class="kw">and </span><br/><a NAME="E6:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E57">A68</font></span>: 
 for <font color="Olive" title="b1">PZ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font> holds <br/>( <font color="Olive" title="b1">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Maroon" title="c14">PX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">PZ</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E18:37_5_1_1"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="txt" href="#E19:37_5_1_1"><span class="lab"><font color="Green" title="E53">A61</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">Pb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E58">A69</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span><span class="kw">set </span><font color="Maroon" title="c17">PS1</font> = <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">C1</font> = <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E9:37_5_1_1_7_1_1"/>
<font color="Maroon" title="c15">Pmax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_5_1_1"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A67</font></span></a></span>;<br/><a NAME="E10:37_5_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A70</font></span>: 
<font color="Maroon" title="c15">Pmax</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E11:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E60">A71</font></span>: 
<font color="Maroon" title="c15">Pmax</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E18:37_5_1_1"><span class="lab"><font color="Green" title="E52">A60</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A67</font></span></a></span>;<br/><a NAME="E12:37_5_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A72</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> or <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> or <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> or <font color="Maroon" title="c10">h</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></span>;<br/><a NAME="E13:37_5_1_1_7_1_1"/>
<font color="Maroon" title="c16">Pb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Pmax</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</font> or <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_1_2"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</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="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font></span><br/>

<span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_1_2"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A69</font></span></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="E14:37_5_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">Pb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E60">A71</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:37_5_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A73</font></span>: 
<font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><div><span class="lab"><font color="Green" title="E63">A74</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">P3</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="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_1_3"/><span class="lab"><font color="Green" title="E64">A75</font></span>: 
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_1_3_1"/>
( <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_1_3"><span class="lab"><font color="Green" title="E64">A75</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_3_1_1"/>
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_1_3_1_1"/>
<font color="Maroon" title="c19">P3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_3_1_2"/>
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_1_3_1_2"/>
<font color="Maroon" title="c19">P3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E17:37_5_1_1_7_1_1"/>
<font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E18:37_5_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A76</font></span>: 
<font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E19:37_5_1_1_7_1_1"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E20:37_5_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A77</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E21:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E66">A78</font></span>: 
 ex <font color="Olive" title="b1">hx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font> &amp; <font color="Olive" title="b1">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hx</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_5_1_1"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A67</font></span></a></span>;<br/><a NAME="E22:37_5_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A79</font></span>: 
<font color="Maroon" title="c15">Pmax</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E10:37_5_1_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E10:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E12:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E61">A72</font></span></a>, <a class="ref" href="taxonom2.html#T19" target="_self" title="TAXONOM2:th.19">Th19</a></span>;<br/><div><span class="lab"><font color="Green" title="E68">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">P3</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"> (  not <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_1_4"/><span class="lab"><font color="Green" title="E69">A81</font></span>: 
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_1_4_1"/>
( <font color="Maroon" title="c11">Ca</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c11">Ca</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_4_1_1"/>
<font color="Maroon" title="c11">Ca</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_5_1_1"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E6:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E57">A68</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1_4"><span class="lab"><font color="Green" title="E69">A81</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_1_4_1_1"/>
( <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E67">A79</font></span></a>, <a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_4_1_2"/><span class="lab"><font color="Green" title="E70">A82</font></span>: 
<font color="Maroon" title="c11">Ca</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_1_4_1_2_1_1"/>
( <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> or <font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_5_1_1"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1_4"><span class="lab"><font color="Green" title="E69">A81</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A83</font></span>: 
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</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="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c20">Pmin</font>, <font color="Maroon" title="c21">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A84</font></span>: 
<font color="Maroon" title="c20">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font>
 <span class="kw">and </span><br/><a NAME="E4:37_5_1_1_7_1_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c21">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font>
 <span class="kw">and </span><br/><a NAME="E5:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A85</font></span>: 
 for <font color="Olive" title="b1">PZ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> holds <br/>( <font color="Olive" title="b1">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">PY</font> &amp; <font color="Maroon" title="c20">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">PZ</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E17:37_5_1_1"><span class="lab"><font color="Green" title="E51">A56</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1_4_1_2"><span class="lab"><font color="Green" title="E70">A82</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E74">A86</font></span>: 
<font color="Maroon" title="c20">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A83</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A85</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E75">A87</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c22">hx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A88</font></span>: 
<font color="Maroon" title="c22">hx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Pmin</font>
 <span class="kw">and </span><br/><a NAME="E3:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A89</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">hx</font>
 <span class="kw">and </span><br/><a NAME="E4:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A90</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">hx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A84</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E5:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">hz</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E79">A91</font></span>: 
<font color="Maroon" title="c23">hz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font>
 <span class="kw">and </span><br/><a NAME="E8:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E80">A92</font></span>: 
<font color="Maroon" title="c22">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">hz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E76">A88</font></span></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">hy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E81">A93</font></span>: 
<font color="Maroon" title="c24">hy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Pmax</font>
 <span class="kw">and </span><br/><a NAME="E11:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E82">A94</font></span>: 
<font color="Maroon" title="c24">hy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">and </span><br/><a NAME="E12:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">hy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_5_1_1"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A67</font></span></a></span>;<br/><a NAME="E13:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E83">A95</font></span>: 
 not <font color="Maroon" title="c24">hy</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E10:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E81">A93</font></span></a>, <a class="ref" href="eqrel_1.html#D4" title="EQREL_1:def.4">EQREL_1:def 4</a></span>;<br/><a NAME="E14:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/>
<font color="Maroon" title="c24">hy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">hx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E77">A89</font></span></a>, <a class="txt" href="#E11:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E82">A94</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">hy</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c23">hz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E80">A92</font></span></a>, <a class="txt" href="#E13:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E83">A95</font></span></a>, <a class="txt" href="taxonom2.html#E29"><span class="lab"><font color="Green" title="E21">Lm4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E16:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">hy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">hz</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E10:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E81">A93</font></span></a>, <a class="txt" href="#E7:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E79">A91</font></span></a>, <a class="ref" href="eqrel_1.html#D4" title="EQREL_1:def.4">EQREL_1:def 4</a></span>;<br/><a NAME="E17:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">hx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E82">A94</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E80">A92</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E18:37_5_1_1_7_1_1_4_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E77">A89</font></span></a>, <a class="txt" href="#E4:37_5_1_1_7_1_1_4_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E78">A90</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="E8:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E76">A96</font></span>: 
<font color="Maroon" title="c15">Pmax</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_5_1_1"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A67</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A97</font></span>: 
<font color="Maroon" title="c15">Pmax</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E10:37_5_1_1_7_1_1_4_1_2_1_1_1"/>
<font color="Maroon" title="c20">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A84</font></span></a></span>;<br/><a NAME="E11:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A98</font></span>: 
<font color="Maroon" title="c20">Pmin</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E12:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E79">A99</font></span>: 
 ex <font color="Olive" title="b1">hw</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">hw</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Pmin</font> &amp; <font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">hw</font> &amp; <font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">hw</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A84</font></span></a></span>;<br/><a NAME="E13:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E80">A100</font></span>: 
<font color="Maroon" title="c20">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_5_1_1"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A84</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E14:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">Pmin</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E15:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">Pmax</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">Pmin</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E13:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A100</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A96</font></span></a>, <a class="txt" href="#E9:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A97</font></span></a>, <a class="txt" href="#E7:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A87</font></span></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</a></span>;<br/><a NAME="E16:37_5_1_1_7_1_1_4_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">Pmin</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E10:37_5_1_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E10:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E12:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E61">A72</font></span></a>, <a class="txt" href="#E21:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E66">A78</font></span></a>, <a class="txt" href="#E11:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A98</font></span></a>, <a class="txt" href="#E12:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A99</font></span></a>, <a class="ref" href="taxonom2.html#T19" target="_self" title="TAXONOM2:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E17:37_5_1_1_7_1_1_4_1_2_1_1_1"/>
( <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_5_1_1_7_1_1_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A86</font></span></a>, <a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_4_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_4_1_2_1_1_2"/>
<font color="Maroon" title="c19">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Cb</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="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_1_4_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">Pmax</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E57">A68</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_1_4_1_2_1_1_2"/>
( <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E67">A79</font></span></a>, <a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_1_4_1_2"/>
( <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P3</font> or <font color="Maroon" title="c19">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E69">A101</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">P1</font>, <font color="Maroon" title="c20">P2</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="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_5_1_1_7_1_1_5"/><span class="lab"><font color="Green" title="E70">A102</font></span>: 
<font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:37_5_1_1_7_1_1_5"/><span class="lab"><font color="Green" title="E71">A103</font></span>: 
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_1_5_1"/>
( <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_1_5"><span class="lab"><font color="Green" title="E70">A102</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_5_1_1"/><span class="lab"><font color="Green" title="E72">A104</font></span>: 
<font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_1_5_1_1_1"/>
( <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_1_5"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A105</font></span>: 
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A106</font></span>: 
<font color="Maroon" title="c20">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E3:37_5_1_1_7_1_1_5_1_1_1_1"/>
<font color="Maroon" title="c19">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_1_5_1_1"><span class="lab"><font color="Green" title="E72">A104</font></span></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:37_5_1_1_7_1_1_5_1_1_1_1"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1_5_1_1"><span class="lab"><font color="Green" title="E72">A104</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E73">A105</font></span></a>, <a class="txt" href="#E2:37_5_1_1_7_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_5_1_1_1_2"/>
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_1_5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_1_5_1_1_1_2"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E68">A80</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_1_5_1_1"><span class="lab"><font color="Green" title="E72">A104</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_5_1_2"/>
<font color="Maroon" title="c19">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A107</font></span>: 
<font color="Maroon" title="c19">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_1_5_1_2_1"/>
( <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_1_5"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_5_1_2_1_1"/>
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_1_5_1_2_1_1"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E68">A80</font></span></a>, <a class="txt" href="#E2:37_5_1_1_7_1_1_5_1_2"><span class="lab"><font color="Green" title="E72">A107</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_1_5_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_1_5_1_2_1_2"/>
<font color="Maroon" title="c20">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_1_5_1_2_1_2"/>
( <font color="Maroon" title="c19">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P2</font> or <font color="Maroon" title="c20">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c19">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_1_5_1_2"><span class="lab"><font color="Green" title="E72">A107</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E25:37_5_1_1_7_1_1"/><span class="lab"><font color="Green" title="E70">A108</font></span>: 
<font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E10:37_5_1_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E10:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E12:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E61">A72</font></span></a>, <a class="txt" href="#E21:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E66">A78</font></span></a>, <a class="ref" href="taxonom2.html#T19" target="_self" title="TAXONOM2:th.19">Th19</a></span>;<br/><a NAME="E26:37_5_1_1_7_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_1_6"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:37_5_1_1_7_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_1_6"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E70">A108</font></span></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1: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/>


</div><span class="kw">end;</span></div><a NAME="E27:37_5_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E28:37_5_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E16:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E63">A74</font></span></a>, <a class="txt" href="#E24:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E69">A101</font></span></a></span>;<br/><a NAME="E29:37_5_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">Pb</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">h</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E30:37_5_1_1_7_1_1"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E65">A77</font></span></a>, <a class="txt" href="#E18:37_5_1_1_7_1_1"><span class="lab"><font color="Green" title="E64">A76</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI: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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E54">A109</font></span>: 
<font color="Maroon" title="c12">Cb</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Pmin</font>, <font color="Maroon" title="c15">PY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E55">A110</font></span>: 
<font color="Maroon" title="c14">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font>
 <span class="kw">and </span><br/><a NAME="E4:37_5_1_1_7_1_2"/>
<font color="Maroon" title="c15">PY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font>
 <span class="kw">and </span><br/><a NAME="E5:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E56">A111</font></span>: 
 for <font color="Olive" title="b1">PZ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">PZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Ca</font> holds <br/>( <font color="Olive" title="b1">PZ</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c15">PY</font> &amp; <font color="Maroon" title="c14">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">PZ</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E24:37"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E15:37_5_1_1"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="txt" href="#E17:37_5_1_1"><span class="lab"><font color="Green" title="E51">A56</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">hw</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E57">A112</font></span>: 
<font color="Maroon" title="c16">hw</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Pmin</font>
 <span class="kw">and </span><br/><a NAME="E8:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E58">A113</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">hw</font>
 <span class="kw">and </span><br/><a NAME="E9:37_5_1_1_7_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">hw</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_5_1_1"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E55">A110</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">hw</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">PT</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E59">A114</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">PS</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span><span class="kw">set </span><font color="Maroon" title="c18">PS1</font> = <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">C1</font> = <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E12:37_5_1_1_7_1_2"/>
<font color="Maroon" title="c17">PT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">PS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</font> or <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">PS</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_2_1"/>
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</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="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">PS</font></span><br/>

<span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_2_1"/>
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">PS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E59">A114</font></span></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="E13:37_5_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A115</font></span>: 
<font color="Maroon" title="c17">PT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_5_1_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E14:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E61">A116</font></span>: 
<font color="Maroon" title="c14">Pmin</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E55">A110</font></span></a></span>;<br/><a NAME="E15:37_5_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E16:37_5_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A117</font></span>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E60">A115</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><div><span class="lab"><font color="Green" title="E63">A118</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">P3</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="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_2_2"/><span class="lab"><font color="Green" title="E64">A119</font></span>: 
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_2_2_1"/>
( <font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_2_2"><span class="lab"><font color="Green" title="E64">A119</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_2_1_1"/>
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_2_2_1_1"/>
<font color="Maroon" title="c20">P3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_2_1_2"/>
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_2_2_1_2"/>
<font color="Maroon" title="c20">P3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E62">A117</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E18:37_5_1_1_7_1_2"/>
<font color="Maroon" title="c14">Pmin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_5_1_1"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="txt" href="#E3:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E55">A110</font></span></a></span>;<br/><a NAME="E19:37_5_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A120</font></span>: 
<font color="Maroon" title="c14">Pmin</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E20:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E65">A121</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">PS</font> or <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">hw</font> or <font color="Maroon" title="c16">hw</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font> or <font color="Maroon" title="c16">hw</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E9:37_5_1_1"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E14:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E61">A116</font></span></a>, <a class="txt" href="#E7:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E57">A112</font></span></a>, <a class="ref" href="taxonom2.html#D3" target="_self" title="TAXONOM2:def.3">Def3</a></span>;<br/><a NAME="E21:37_5_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A122</font></span>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c14">Pmin</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_5_1_1"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="txt" href="#E10:37_5_1_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E19:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E64">A120</font></span></a>, <a class="txt" href="#E7:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E57">A112</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E58">A113</font></span></a>, <a class="txt" href="#E11:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E59">A114</font></span></a>, <a class="ref" href="taxonom2.html#T18" target="_self" title="TAXONOM2:th.18">Th18</a></span>;<br/><div><span class="lab"><font color="Green" title="E67">A123</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">P3</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"> (  not <font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P3</font> or <font color="Maroon" title="c20">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_2_3"/>
<font color="Maroon" title="c20">P3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</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="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P3</font> or <font color="Maroon" title="c20">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )</span><br/><a NAME="E2:37_5_1_1_7_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">Pmin</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_5_1_1"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E54">A109</font></span></a>, <a class="txt" href="#E5:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E56">A111</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_2_3"/>
( <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P3</font> or <font color="Maroon" title="c20">P3</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E66">A122</font></span></a>, <a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_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><div><span class="lab"><font color="Green" title="E68">A124</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">P1</font>, <font color="Maroon" title="c21">P2</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="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_5_1_1_7_1_2_4"/><span class="lab"><font color="Green" title="E69">A125</font></span>: 
<font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E2:37_5_1_1_7_1_2_4"/><span class="lab"><font color="Green" title="E70">A126</font></span>: 
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_2_4_1"/>
( <font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_2_4"><span class="lab"><font color="Green" title="E69">A125</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_4_1_1"/><span class="lab"><font color="Green" title="E71">A127</font></span>: 
<font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_2_4_1_1_1"/>
( <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_2_4"><span class="lab"><font color="Green" title="E70">A126</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_4_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A128</font></span>: 
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_2_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A129</font></span>: 
<font color="Maroon" title="c21">P2</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><a NAME="E3:37_5_1_1_7_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c20">P1</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_5_1_1_7_1_2_4_1_1"><span class="lab"><font color="Green" title="E71">A127</font></span></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:37_5_1_1_7_1_2_4_1_1_1_1"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_2_4_1_1"><span class="lab"><font color="Green" title="E71">A127</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E72">A128</font></span></a>, <a class="txt" href="#E2:37_5_1_1_7_1_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E73">A129</font></span></a>, <a class="ref" href="taxonom1.html#D1" title="TAXONOM1:def.1">TAXONOM1:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_4_1_1_1_2"/>
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_2_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_2_4_1_1_1_2"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E67">A123</font></span></a>, <a class="txt" href="#E1:37_5_1_1_7_1_2_4_1_1"><span class="lab"><font color="Green" title="E71">A127</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_4_1_2"/>
<font color="Maroon" title="c20">P1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:37_5_1_1_7_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A130</font></span>: 
<font color="Maroon" title="c20">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_5_1_1_7_1_2_4_1_2_1"/>
( <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_2_4"><span class="lab"><font color="Green" title="E70">A126</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_4_1_2_1_1"/>
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_2_4_1_2_1_1"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E67">A123</font></span></a>, <a class="txt" href="#E2:37_5_1_1_7_1_2_4_1_2"><span class="lab"><font color="Green" title="E71">A130</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_7_1_2_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_1_7_1_2_4_1_2_1_2"/>
<font color="Maroon" title="c21">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>1</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_1_7_1_2_4_1_2_1_2"/>
( <font color="Maroon" title="c20">P1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c21">P2</font> or <font color="Maroon" title="c21">P2</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c20">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_5_1_1_7_1_2_4_1_2"><span class="lab"><font color="Green" title="E71">A130</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E24:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E69">A131</font></span>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_5_1_1"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="txt" href="#E10:37_5_1_1"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E19:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E64">A120</font></span></a>, <a class="txt" href="#E7:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E57">A112</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E58">A113</font></span></a>, <a class="txt" href="#E11:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E59">A114</font></span></a>, <a class="txt" href="#E20:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E65">A121</font></span></a>, <a class="ref" href="taxonom2.html#T18" target="_self" title="TAXONOM2:th.18">Th18</a></span>;<br/><a NAME="E25:37_5_1_1_7_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_7_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1_1_7_1_2_5"/>
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:37_5_1_1_7_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_5_1_1_7_1_2_5"/>
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E69">A131</font></span></a>, <a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1: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/>


</div><span class="kw">end;</span></div><a NAME="E26:37_5_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E27:37_5_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">RL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E17:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E63">A118</font></span></a>, <a class="txt" href="#E23:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E68">A124</font></span></a></span>;<br/><a NAME="E28:37_5_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A132</font></span>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E29:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E71">A133</font></span>: 
<font color="Maroon" title="c17">PT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E30:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E72">A134</font></span>: 
<font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E31:37_5_1_1_7_1_2"/><span class="lab"><font color="Green" title="E73">A135</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">PT</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">Pmin</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">hw</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E32:37_5_1_1_7_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">PT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_5_1_1"><span class="lab"><font color="Green" title="E44">A40</font></span></a>, <a class="txt" href="#E8:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E58">A113</font></span></a>, <a class="txt" href="#E11:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E59">A114</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E33:37_5_1_1_7_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E71">A133</font></span></a>, <a class="txt" href="#E31:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E73">A135</font></span></a>, <a class="txt" href="#E30:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E72">A134</font></span></a>, <a class="txt" href="#E28:37_5_1_1_7_1_2"><span class="lab"><font color="Green" title="E70">A132</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI: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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:37_5_1_1"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_5_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</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="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:37_5_1_2"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font>
 ; <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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c9">C</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="c9">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )</span><br/>

<a NAME="E26:37"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> or <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_6"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</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="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37_6"/><span class="lab"><font color="Green" title="E39">A136</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font>
 <span class="kw">and </span><br/><a NAME="E4:37_6"/><span class="lab"><font color="Green" title="E40">A137</font></span>: 
<font color="Maroon" title="c11">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:37_6"/>
<font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E20:37"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E4:37_6"><span class="lab"><font color="Green" title="E40">A137</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:37_6"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_6"><span class="lab"><font color="Green" title="E39">A136</font></span></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="E27:37"/>
( <font color="Maroon" title="c9">C</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E25:37"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
