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<div class="add">

<b>let </b><font color="Maroon">C</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">E</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">D</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">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">E</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,<font color="Maroon">E</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">H</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d1</font>,<font color="Olive">d2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">d2</font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>] <b>means </b><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>;<br/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">f</font></span>)</span> = 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H</font>
<b>by </b><i>, , <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>
;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">C</font>]
 
;<br/>
<a NAME="E7:18"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">B'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font><br/> for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font>] &amp; not <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B'</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">b</font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E61</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>thus </b><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K5">\/</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:18"/>
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2">SETWISEO:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E9:18"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


</div>
