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<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel_3.html#V5">reflexive-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<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"><font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>









<b>assume </b><a NAME="E1:48"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>assume </b><a NAME="E4:48"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font>, <font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span><b> such that </b><br/><a NAME="E9:48"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">g</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">h</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E10:48"/>
<font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>

<b>thus </b><a NAME="E11:48"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , , <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


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