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

<b>let </b><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> ;<br/><b>let </b><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#V2" title="WAYBEL_0:attr.2">filtered</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>;<br/>



<b>set </b><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/>
<b>set </b><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/>
<b>reconsider </b><font color="Maroon" title="c6">C1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c7">C2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font> as  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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">D'</font> = <font color="Maroon" title="c3">D</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c6">C1</font>,<font color="Maroon" title="c7">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="yellow_3.html#D2" target="_self" title="YELLOW_3:def.2">Def2</a></i>;<br/>
<a NAME="E3:82"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T1" target="_self" title="YELLOW_3:th.1">Th1</a></i>;<br/>
<a NAME="E4:82"/>
not  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c8">D'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">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> ;<br/>
<a NAME="E6:82"/>
not  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c8">D'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">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> ;<br/>
<b>let </b><font color="Maroon" title="c11">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E8:82"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:82"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">D1</font> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D2</font> &amp; <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:82"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> :  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> st <br/>( <font color="Olive" title="b1">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T15" title="WAYBEL_0:th.15">WAYBEL_0:15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:82"/><i><font color="Green" title="E27">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">y</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>
 <b>by </b><i><a class="txt" href="yellow_3.html#E10:82"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:82"/><i><font color="Green" title="E28">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c13">e</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>
 <b>by </b><i><a class="txt" href="yellow_3.html#E10:82"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c17">d</font> = <font color="Maroon" title="c12">d</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c9">D1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E13:82"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="yellow_3.html#E15:82"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="funct_5.html#T4" title="FUNCT_5:th.4">FUNCT_5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c19">e</font> = <font color="Maroon" title="c13">e</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c10">D2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E13:82"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="yellow_3.html#E15:82"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="funct_5.html#T4" title="FUNCT_5:th.4">FUNCT_5:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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><b> such that </b><br/><a NAME="E19:82"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c18">y</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> <font color="Maroon" title="c20">z</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c19">e</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> <font color="Maroon" title="c20">z</font> )
 <b>by </b><i><a class="txt" href="yellow_3.html#E13:82"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="yellow_3.html#E15:82"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">z1</font>, <font color="Maroon" title="c22">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:82"/><i><font color="Green" title="E30">A7</font></i>: 
( <font color="Maroon" title="c21">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">D1</font> &amp; <font color="Maroon" title="c22">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">D2</font> &amp; <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">z1</font>,<font color="Maroon" title="c22">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="yellow_3.html#E3:82"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow_3.html#E19:82"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">z1</font> = <font color="Maroon" title="c21">z1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c9">D1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E21:82"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">z2</font> = <font color="Maroon" title="c22">z2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c10">D2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E21:82"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/>
<a NAME="E24:82"/>
( <font color="Maroon" title="c17">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c23">z1</font> &amp; <font color="Maroon" title="c19">e</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c24">z2</font> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E19:82"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="yellow_3.html#E21:82"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></i>;<br/>
<a NAME="E25:82"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c19">e</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="c23">z1</font>,<font color="Maroon" title="c24">z2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E26:82"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">D</font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E10:82"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="yellow_3.html#E11:82"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="yellow_3.html#E19:82"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="yellow_3.html#E21:82"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/>


</div>
