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<b>let </b><font color="Maroon">F</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>, <font color="Maroon">G</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> implies <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> implies <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0 ) )
 <b>and </b><br/><a NAME="E2:28_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> implies <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> implies <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/>

<a NAME="E3:28_1_2"/><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E4:28_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> holds <br/><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<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> ;<br/>

<b>assume </b><a NAME="E1:28_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 ;<br/>

<a NAME="E2:28_1_2_1"/><b>then </b>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:28_1_2_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <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#R1">=</a> 1 ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <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#R1">=</a> 0 ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <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#R1">=</a> 1 ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> implies <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <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#R1">=</a> 0 ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:28_1_2_1"/>
<font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#T6">FUNCT_3:6</a></i>;<br/>


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