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<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">S</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</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">g</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="realset1.html#K1">||</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:1"/><i><font color="Green">E15</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">S</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:1"/>
( <font color="Maroon">f</font> is <a href="binop_1.html#V1">commutative</a> implies <font color="Maroon">g</font> is <a href="binop_1.html#V1">commutative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b'</font> = <font color="Maroon">b'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>

<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">a'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:1"/>
( <font color="Maroon">f</font> is <a href="binop_1.html#V3">idempotent</a> implies <font color="Maroon">g</font> is <a href="binop_1.html#V3">idempotent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15">BINOP_1:def 15</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15">BINOP_1:def 15</a><br/>

<b>thus </b><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>

<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span>,<font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">c</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14">BINOP_1:def 14</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14">BINOP_1:def 14</a><br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b'</font> = <font color="Maroon">b'</font>, <font color="Maroon">c'</font> = <font color="Maroon">c'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>

<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>


</div>
