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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="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">K</font> be  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">non-Empty</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">K</font> <a href="waybel_3.html#K3">.</a> <font color="Olive">i</font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">J</font> <a href="waybel_3.html#K3">.</a> <font color="Olive">i</font>
 ;<br/>

<a NAME="E2:49"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">K</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">K</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E3:49_1"/><i><font color="Green">E49</font></i>: 
(  ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> ) &amp;  ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">K</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> ) )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E4:49_1"/>
<font color="Maroon">K</font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">j</font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">J</font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:49_1"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">K</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:49"/><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">K</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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>, , , <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a><br/>

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E7:49"/><i><font color="Green">E93</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="yellow_1.html#K5">product</a> <font color="Maroon">K</font></span>)</span>
 ;<br/>

<a NAME="E8:49"/><b>then </b><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">K</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">K</font></span>)</span> )
 
<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="yellow_1.html#K5">product</a> <font color="Maroon">K</font></span>)</span> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</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/>
<a NAME="E11:49"/><i><font color="Green">E100</font></i>: 
<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="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E1:49_2"/>
( <font color="Maroon">K</font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">i</font> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">J</font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/><b>hence </b><a NAME="E2:49_2"/>
<font color="Maroon">f</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:49"/><b>then </b>
<font color="Maroon">f</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E14:49"/>
<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="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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