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<div class="add">

<b>let </b><font color="Maroon" title="c4">it1</font>, <font color="Maroon" title="c5">it2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="waybel_0.html#L1" title="WAYBEL_0:struct.1">NetStr</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_2"/><i><font color="Green" title="E22">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E2:45_1_2"/><i><font color="Green" title="E23">A3</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">it1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E3:45_1_2"/><i><font color="Green" title="E24">A4</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it1</font> holds <br/> ( <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> iff <font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font> )
 <b>and </b><br/><a NAME="E4:45_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E5:45_1_2"/><i><font color="Green" title="E26">A6</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c5">it2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E6:45_1_2"/><i><font color="Green" title="E27">A7</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">it2</font> holds <br/> ( <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> iff <font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">j</font> )
 ;<br/>

<a NAME="E7:45_1_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">it1</font> <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 <font color="Maroon" title="c5">it2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">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="E1:45_1_2_1_1"/><i><font color="Green" title="E28">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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 <font color="Maroon" title="c4">it1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it1</font> <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="c10">i'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c11">j'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">it2</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:45_1_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="waybel11.html#E4:45_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="waybel11.html#E1:45_1_2_1_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:45_1_2_1_1"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">i'</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:45_1_2"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="waybel11.html#E5:45_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><a NAME="E5:45_1_2_1_1"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">j'</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:45_1_2"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="waybel11.html#E5:45_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><a NAME="E6:45_1_2_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:45_1_2_1_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E7:45_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">j'</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:45_1_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="waybel11.html#E4:45_1_2_1_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="waybel11.html#E5:45_1_2_1_1"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/><a NAME="E8:45_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">j'</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E6:45_1_2"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E9:45_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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 <font color="Maroon" title="c5">it2</font>
 <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="E2:45_1_2_1"/><i><font color="Green" title="E28">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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 <font color="Maroon" title="c5">it2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">it2</font> <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="c10">i'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c11">j'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it1</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:45_1_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="waybel11.html#E4:45_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="waybel11.html#E2:45_1_2_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:45_1_2_1"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">i'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E5:45_1_2"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="waybel11.html#E2:45_1_2"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>
<a NAME="E6:45_1_2_1"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c5">it2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">j'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E5:45_1_2"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="waybel11.html#E2:45_1_2"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>
<a NAME="E7:45_1_2_1"/>
<font color="Maroon" title="c8">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:45_1_2_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:45_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">it1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">j'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E6:45_1_2"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="waybel11.html#E5:45_1_2_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="waybel11.html#E6:45_1_2_1"><i><font color="Green" title="E30">A13</font></i></a></i>;<br/>
<a NAME="E9:45_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">j'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:45_1_2"><i><font color="Green" title="E24">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:45_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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 <font color="Maroon" title="c4">it1</font>
 <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="E8:45_1_2"/>
<font color="Maroon" title="c4">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">it2</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:45_1_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="waybel11.html#E2:45_1_2"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="waybel11.html#E4:45_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="waybel11.html#E5:45_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>


</div>
