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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="orders_2.html#V2">reflexive</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>;<br/>





<a NAME="E1:34"/><i><font color="Green">E27</font></i>: 
 <a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<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"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">x'</font>, <font color="Maroon">ya</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:34_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="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/><a NAME="E3:34_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">xa</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">ya</font>
 <b>by </b><i><a class="ref" href="waybel10.html#T12">WAYBEL10:12</a></i>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ya</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/><b>thus </b><a NAME="E6:34_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></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">T</font>
 <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="E5:34"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></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">T</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">xa</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>, <font color="Maroon">b</font> = <font color="Maroon">ya</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>thus </b><a NAME="E4:34_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ya</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><a NAME="E5:34_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:34_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:34"/><b>then </b>
<font color="Maroon">xa</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">ya</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<a NAME="E8:34"/><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="waybel10.html#T12">WAYBEL10:12</a></i>;<br/>
<b>hence </b><a NAME="E9:34"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


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