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

<span class="kw">let </span><font color="Maroon" title="c3">it1</font>, <font color="Maroon" title="c4">it2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <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="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> iff <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font> ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> iff <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font> ) ) implies <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font> ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57_2_2_1"/>
 not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">and </span><br/><a NAME="E2:57_2_2_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> iff <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 <span class="kw">and </span><br/><a NAME="E3:57_2_2_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> iff <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> iff <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_2_2_1_1_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x9</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E3:57_2_2_1_1_1"/>
<font color="Maroon" title="c6">x9</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2_2_1_1_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E2:57_2_2_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E3:57_2_2_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:57_2_2_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2_2_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E3:57_2_2_1"><span class="lab"><font color="Green" title="E29">A11</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="E2:57_2_2_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> iff <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</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><span class="kw">hence </span><a NAME="E5:57_2_2_1"/>
<font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI: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">hence </span><a NAME="E2:57_2_2"/>
( (  not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">it1</font> iff <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font> ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">it2</font> iff <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font> ) ) implies <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font> ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</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>
