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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</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">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">S1</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:33"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>( <font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a> &amp;  <a href="funct_5.html#K1">proj1</a> <font color="Olive">x</font> is   <a href="relat_1.html#NM1">Relation</a> )
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <font color="Olive">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel27.html#D2" target="_self">WAYBEL27:def 2</a><br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font>, <font color="Maroon">b</font> = <font color="Maroon">g</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> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span></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">Fa</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>, <font color="Maroon">Fb</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>assume </b><a NAME="E5:33"/>
<font color="Maroon">f</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E6:33"/><b>then </b><i><font color="Green">E38</font></i>: 
<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_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<a NAME="E9:33"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:33"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><b>reconsider </b><font color="Maroon">xy</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><a NAME="E2:33_1_1"/>
( <font color="Maroon">Fa</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Fb</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Fa</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Fb</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E3:33_1_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fa</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:33_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Fa</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_5.html#T38">FUNCT_5:38</a></i>;<br/><a NAME="E5:33_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Fa</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">xy</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">xy</font> )
 ;<br/><b>hence </b><a NAME="E6:33_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Fa</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">Fb</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:33_1"/>
<font color="Maroon">Fa</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Fb</font> <a href="waybel27.html#K1" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E12:33"/><b>then </b>
( <font color="Maroon">Fa</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Fb</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:33"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


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