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

<b>set </b><font color="Maroon" title="c3">IT</font> = <font color="Maroon" title="c2">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" title="LATTICE3:def.11">LATTICE3:def 11</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="yellow_1.html#E38"><i><font color="Green" title="E16">Lm5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c5">y</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="yellow_1.html#E38"><i><font color="Green" title="E16">Lm5</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">xa</font>, <font color="Olive" title="b2">ya</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b1">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">ya</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xa</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">ya</font> );<br/>
<a NAME="E3:56_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">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 <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">xa</font> = <font color="Maroon" title="c6">x'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c10">ya</font> = <font color="Maroon" title="c7">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>take </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">xa</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">ya</font>;<br/>

<b>thus </b><a NAME="E3:56_1_1_1"/>
<font color="Maroon" title="c11">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 <font color="Maroon" title="c2">Y</font>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c9">xa</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">ya</font>
;<br/>

<b>thus </b><a NAME="E4:56_1_1_1"/>
( <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c10">ya</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xa</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">ya</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E5:56_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="yellow_1.html#E3:56_1_1"><i><font color="Green" title="E17">A1</font></i></a>);<br/></i>
<a NAME="E6:56_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T28" target="_self" title="YELLOW_1:th.28">Th28</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">z</font>
;<br/>

<a NAME="E8:56_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c9">z</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ;<br/>
<a NAME="E2:56_1_1_2"/>
<font color="Maroon" title="c11">z1</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:56_1_1_2"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c11">z1</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>
<a NAME="E4:56_1_1_2"/>
 ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z1</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b4">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b5">xi</font>, <font color="Olive" title="b6">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">R</font> st <br/>( <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b6">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">f</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">x'</font>
;<br/>

<b>thus </b><a NAME="E1:56_1_1_2_1"/>
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z1</font> &amp; <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x1</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:56_1_1_2_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">R</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c13">i</font> as    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<a NAME="E5:56_1_1_2_1"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c14">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">xi</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c16">yi</font> = <font color="Maroon" title="c6">x'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">R</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">xi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c16">yi</font>
;<br/>

<b>consider </b><font color="Maroon" title="c17">xa</font>, <font color="Maroon" title="c18">ya</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E8:56_1_1_2_1"/><i><font color="Green" title="E22">A6</font></i>: 
( <font color="Maroon" title="c17">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c18">ya</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xa</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c18">ya</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E5:56_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:56_1_1_2_1"/>
( <font color="Maroon" title="c14">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c15">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c16">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <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="yellow_1.html#E5:56_1_1_2_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="yellow_1.html#E8:56_1_1_2_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:56_1_1_2"/>
<font color="Maroon" title="c9">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E3:56_1_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:56_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c9">z</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ;<br/>
<a NAME="E2:56_1_1_3"/>
<font color="Maroon" title="c11">z1</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:56_1_1_3"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c11">z1</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>
<a NAME="E4:56_1_1_3"/>
 ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z1</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b4">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b5">xi</font>, <font color="Olive" title="b6">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">R</font> st <br/>( <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b6">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c8">f</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">y'</font>
;<br/>

<b>thus </b><a NAME="E1:56_1_1_3_1"/>
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z1</font> &amp; <font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:56_1_1_3_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">R</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c13">i</font> as    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<a NAME="E5:56_1_1_3_1"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c14">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">xi</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c16">yi</font> = <font color="Maroon" title="c7">y'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">R</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">xi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c16">yi</font>
;<br/>

<b>consider </b><font color="Maroon" title="c17">xa</font>, <font color="Maroon" title="c18">ya</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E8:56_1_1_3_1"/><i><font color="Green" title="E23">A10</font></i>: 
( <font color="Maroon" title="c17">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c18">ya</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xa</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c18">ya</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E5:56_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:56_1_1_3_1"/>
( <font color="Maroon" title="c14">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c15">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c16">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <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="yellow_1.html#E5:56_1_1_3_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="yellow_1.html#E8:56_1_1_3_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:56_1_1_3"/>
<font color="Maroon" title="c9">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E3:56_1_1_3"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:56_1_1"/>
for <font color="Olive" title="b1">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> holds <br/><font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:56_1_1_4"/><i><font color="Green" title="E21">A11</font></i>: 
( <font color="Maroon" title="c10">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c10">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> )
 ;<br/>

<a NAME="E2:56_1_1_4"/>
<font color="Maroon" title="c10">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">z2</font> = <font color="Maroon" title="c10">z'</font>, <font color="Maroon" title="c12">z3</font> = <font color="Maroon" title="c9">z</font>, <font color="Maroon" title="c13">z4</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c14">z5</font> = <font color="Maroon" title="c4">x</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">J</font> = <font color="Maroon" title="c11">z2</font>, <font color="Maroon" title="c16">K</font> = <font color="Maroon" title="c12">z3</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="yellow_1.html#E38"><i><font color="Green" title="E16">Lm5</font></i></a></i>;<br/>
<a NAME="E3:56_1_1_4_1"/>
<font color="Maroon" title="c10">z'</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:56_1_1_4_1"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c11">z2</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">f1</font>, <font color="Maroon" title="c18">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:56_1_1_4_1"/><i><font color="Green" title="E23">A13</font></i>: 
( <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c18">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z5</font> &amp; ( 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">X</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></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="c17">f1</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="c18">g1</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="yellow_1.html#E1:56_1_1_4"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">f2</font>, <font color="Maroon" title="c20">g2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:56_1_1_4_1"/><i><font color="Green" title="E24">A14</font></i>: 
( <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z4</font> &amp; ( 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">X</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">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></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="c19">f2</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="c20">g2</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="yellow_1.html#E1:56_1_1_4"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="yellow_1.html#E4:56_1_1_4_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>
<a NAME="E9:56_1_1_4_1"/>
 ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z3</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b4">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b5">xi</font>, <font color="Olive" title="b6">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">R</font> st <br/>( <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b6">yi</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c15">J</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c16">K</font>
;<br/>

<b>thus </b><a NAME="E1:56_1_1_4_1_1"/>
( <font color="Maroon" title="c15">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c16">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z3</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c21">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:56_1_1_4_1_1"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c21">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c23">R</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c22">i</font> as    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<a NAME="E5:56_1_1_4_1_1"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c23">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">xi</font> = <font color="Maroon" title="c15">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font>, <font color="Maroon" title="c25">yi</font> = <font color="Maroon" title="c16">K</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c23">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c23">R</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c24">xi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c25">yi</font>
;<br/>

<b>consider </b><font color="Maroon" title="c26">xa</font>, <font color="Maroon" title="c27">ya</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E8:56_1_1_4_1_1"/><i><font color="Green" title="E26">A16</font></i>: 
( <font color="Maroon" title="c26">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">ya</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">xa</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c27">ya</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E5:56_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">R1</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c29">xi1</font>, <font color="Maroon" title="c30">yi1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c28">R1</font><b> such that </b><br/><a NAME="E10:56_1_1_4_1_1"/><i><font color="Green" title="E27">A17</font></i>: 
( <font color="Maroon" title="c28">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c29">xi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c30">yi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c29">xi1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c30">yi1</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E6:56_1_1_4_1"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">R2</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c32">xi2</font>, <font color="Maroon" title="c33">yi2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c31">R2</font><b> such that </b><br/><a NAME="E12:56_1_1_4_1_1"/><i><font color="Green" title="E28">A18</font></i>: 
( <font color="Maroon" title="c31">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c32">xi2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c33">yi2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c32">xi2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c33">yi2</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E8:56_1_1_4_1"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/>
<b>thus </b><a NAME="E13:56_1_1_4_1_1"/>
( <font color="Maroon" title="c23">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c24">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c25">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c24">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c25">yi</font> )
 <b>by </b><i><a class="txt" href="yellow_1.html#E6:56_1_1_4_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="yellow_1.html#E8:56_1_1_4_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="yellow_1.html#E5:56_1_1_4_1_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="yellow_1.html#E8:56_1_1_4_1_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="yellow_1.html#E10:56_1_1_4_1_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="yellow_1.html#E12:56_1_1_4_1_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:56_1_1_4_1"/>
<font color="Maroon" title="c10">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E4:56_1_1_4_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="yellow_1.html#D4" target="_self" title="YELLOW_1:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_1_1_4"/>
<font color="Maroon" title="c10">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:56_1_1"/>
( <font color="Maroon" title="c9">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c9">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">X</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> ) ) )
 <b>by </b><i><a class="txt" href="yellow_1.html#E8:56_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="yellow_1.html#E9:56_1_1"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>


</div>
