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

<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds <br/> (  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Olive" title="b1">D</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  &amp;  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Olive" title="b1">D</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  &amp;  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">D1</font> =  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D2</font> =  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">d1</font> =  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#T21" target="_self" title="YELLOW_3:th.21">Th21</a></span>;<br/>
<a NAME="E2:52"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#D2" target="_self" title="YELLOW_3:def.2">Def2</a></span>;<br/>
<a NAME="E3:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#T1" target="_self" title="YELLOW_3:th.1">Th1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:52"/>
 <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">d2</font> =  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#T21" target="_self" title="YELLOW_3:th.21">Th21</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> or  not <font color="Maroon" title="c9">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:52_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">and </span><br/><a NAME="E3:52_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">q1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:52_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c10">q1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">d2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E5:52_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:52_1"/>
<font color="Maroon" title="c11">q1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">q1</font>
 
;<br/>
<a NAME="E8:52_1"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c11">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></span>;<br/>
<a NAME="E9:52_1"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c11">q1</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:52_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> or  not <font color="Maroon" title="c8">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:52"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">and </span><br/><a NAME="E6:52"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">q1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:52"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q1</font> = <font color="Maroon" title="c9">q1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">d1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E8:52"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:52"/>
<font color="Maroon" title="c10">q1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">q1</font>
 
;<br/>
<a NAME="E11:52"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c7">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c8">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></span>;<br/>
<a NAME="E12:52"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c8">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:52"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
