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

<b>let </b><font color="Maroon" title="c1">M</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> ;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>



<i><font color="Green" title="E14">A1</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:30"/><i><font color="Green" title="E15">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:30_2"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">R1</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:30_2"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>and </b><br/><a NAME="E5:30_2"/><i><font color="Green" title="E18">A5</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R1</font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">R2</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E7:30_2"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c5">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>and </b><br/><a NAME="E8:30_2"/><i><font color="Green" title="E20">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R2</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:30_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R1</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="waybel25.html#E2:30_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel25.html#E4:30_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E10">Lm1</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R2</font>
<b>by </b><i><a class="txt" href="waybel25.html#E2:30_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel25.html#E7:30_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:30_2"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E5:30_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="waybel25.html#E8:30_2"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E16">A8</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E10">Lm1</font></i></a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#D3" title="WAYBEL18:def.3">WAYBEL18:def 3</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E1:30"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="waybel25.html#E2:30"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
;<br/>
<a NAME="E4:30"/><i><font color="Green" title="E17">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E10">Lm1</font></i></a></i>;<br/>
<a NAME="E5:30"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_4_1"/><i><font color="Green" title="E18">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:30_4_1"/><b>then </b><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">xo</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">yo</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E1:30_4_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">p1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c8">p2</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E2:30_4_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E10">Lm1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">xp</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c10">yp</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E3:30"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="waybel25.html#E2:30_4_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><a NAME="E6:30_4_1"/>
<font color="Maroon" title="c5">xo</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">yo</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:30_4_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">Yp</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:30_4_1"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c11">Yp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">p2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E9:30_4_1"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c7">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">Yp</font>
 <b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/><a NAME="E10:30_4_1"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c9">xp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E3:30"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="waybel25.html#E1:30_4_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:30_4_1"/><i><font color="Green" title="E23">A15</font></i>: 
<font color="Maroon" title="c9">xp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <b>and </b><br/><a NAME="E13:30_4_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E14:30_4_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E15:30_4_1"/>
<font color="Maroon" title="c10">yp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E3:30"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="waybel25.html#E1:30_4_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E17:30_4_1"/><i><font color="Green" title="E24">A16</font></i>: 
<font color="Maroon" title="c10">yp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font>
 <b>and </b><br/><a NAME="E18:30_4_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E19:30_4_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E20:30_4_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_4_1_1"/><i><font color="Green" title="E25">A17</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E3:30_4_1_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c7">p1</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p2</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c15">j</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:30_4_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="waybel25.html#E9:30_4_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="yellow14.html#T31" title="YELLOW14:th.31">YELLOW14:31</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">t1</font> = <font color="Maroon" title="c7">p1</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c15">j</font>;<br/>
<b>set </b><font color="Maroon" title="c17">t2</font> = <font color="Maroon" title="c8">p2</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c15">j</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">xi</font> = <font color="Maroon" title="c7">p1</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c15">j</font>, <font color="Maroon" title="c19">yi</font> = <font color="Maroon" title="c8">p2</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c15">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E10">Lm1</font></i></a></i>;<br/>
<b>take </b>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c18">xi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c19">yi</font>
;<br/>

<b>thus </b><a NAME="E5:30_4_1_1"/>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:30_4_1_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>

<b>thus </b><a NAME="E6:30_4_1_1"/>
<font color="Maroon" title="c18">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E12:30_4_1"><i><font color="Green" title="E23">A15</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:30_4_1_1"/>
<font color="Maroon" title="c19">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E17:30_4_1"><i><font color="Green" title="E24">A16</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:30_4_1_1"/>
<font color="Maroon" title="c18">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">yi</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E3:30_4_1_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:30_4_1"/><b>then </b>
<font color="Maroon" title="c9">xp</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">yp</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E10:30_4_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="waybel25.html#E12:30_4_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="waybel25.html#E17:30_4_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>hence </b><a NAME="E22:30_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:30_4"/><i><font color="Green" title="E18">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:30_4"/><b>then </b><i><font color="Green" title="E19">A20</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">xp</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">yp</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E2:30_4"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">xo</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c8">yo</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E3:30"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="waybel25.html#E3:30_4"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">p1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c10">p2</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E3:30"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="waybel25.html#E4:30"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="waybel25.html#E3:30_4"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E7:30_4"/><i><font color="Green" title="E20">A21</font></i>: 
<font color="Maroon" title="c5">xp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel25.html#E3:30_4"><i><font color="Green" title="E19">A20</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E8:30_4"/>
<font color="Maroon" title="c5">xp</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">yp</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:30_4"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">f</font>, <font color="Maroon" title="c12">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:30_4"/><i><font color="Green" title="E21">A22</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xp</font>
 <b>and </b><br/><a NAME="E11:30_4"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">yp</font>
 <b>and </b><br/><a NAME="E12:30_4"/><i><font color="Green" title="E23">A24</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <b>by </b><i><a class="txt" href="waybel25.html#E7:30_4"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E13:30_4"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <font color="Maroon" title="c9">p1</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p2</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>consider </b><font color="Maroon" title="c14">R1</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c15">xi</font>, <font color="Maroon" title="c16">yi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">R1</font><b> such that </b><br/><a NAME="E2:30_4_2"/><i><font color="Green" title="E24">A25</font></i>: 
<font color="Maroon" title="c14">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c13">i</font>
 <b>and </b><br/><a NAME="E3:30_4_2"/><i><font color="Green" title="E25">A26</font></i>: 
( <font color="Maroon" title="c15">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c16">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )
 <b>and </b><br/><a NAME="E4:30_4_2"/><i><font color="Green" title="E26">A27</font></i>: 
<font color="Maroon" title="c15">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">yi</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E12:30_4"><i><font color="Green" title="E23">A24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">xi</font> = <font color="Maroon" title="c15">xi</font>, <font color="Maroon" title="c18">yi</font> = <font color="Maroon" title="c16">yi</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E2:30_4_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E6:30_4_2"/>
<font color="Maroon" title="c17">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">yi</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:30_4_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="txt" href="waybel25.html#E4:30_4_2"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:30_4_2"/><b>then </b>
 ex <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">yi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c17">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">Y</font> )
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E8:30_4_2"/>
<font color="Maroon" title="c9">p1</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">p2</font> <a href="yellow14.html#K2" title="YELLOW14:func.2">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel25.html#E10:30_4"><i><font color="Green" title="E21">A22</font></i></a>, <a class="txt" href="waybel25.html#E11:30_4"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="waybel25.html#E3:30_4_2"><i><font color="Green" title="E25">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:30_4"/><b>then </b>
<font color="Maroon" title="c9">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">p2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow14.html#T31" title="YELLOW14:th.31">YELLOW14:31</a></i>;<br/>
<a NAME="E15:30_4"/><b>then </b>
<font color="Maroon" title="c7">xo</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">yo</font>
 
<b>by </b><i><a class="ref" href="waybel25.html#D2" target="_self" title="WAYBEL25:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E16:30_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:30"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> #)
 <b>by </b><i><a class="txt" href="waybel25.html#E3:30"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>


</div>
