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<b>set </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>;<br/>
<b>take </b><font color="Maroon">X</font> =  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p3">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p4">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:36_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_1_1_1"/><i><font color="Green">E41</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="funct_6.html#K1">SubFuncs</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="funct_6.html#D1">FUNCT_6:def 1</a></i>;<br/><a NAME="E7:36_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_6.html#D1">FUNCT_6:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:36_1_1_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/><b>take </b><font color="Maroon">c</font> = <font color="Maroon">c</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>thus </b><a NAME="E10:36_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E2:36_1_1"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 ;<br/>

<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E5:36_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E6:36_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_6.html#K1">SubFuncs</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#D1">FUNCT_6:def 1</a></i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:36_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>


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