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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">M</font> =  <a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span>;<br/>
<a NAME="E1:26"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D10">WAYBEL11:def 10</a></i>;<br/>
<a NAME="E2:26"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D10">WAYBEL11:def 10</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#T1">FUNCT_3:1</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E48</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:26_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:26_1_1"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> &amp; <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i/>;<br/><a NAME="E5:26_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E6:26_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E7:26_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D10">WAYBEL11:def 10</a></i>;<br/><b>hence </b><a NAME="E8:26_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:26_1"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:26_1"/>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E5:26_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D10">WAYBEL11:def 10</a></i>;<br/><a NAME="E6:26_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y'</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E7:26_1"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> &amp; <span class="p1">(<span class="default"><a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i/>;<br/><a NAME="E8:26_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E9:26_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">D</font>
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
 <a href="waybel17.html#K4" target="_self">Net-Str</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3">Net-Str</a> <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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