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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span>)</span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : verum </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:3"/>
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <font color="Maroon" title="c1">V</font> implies  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( ( <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span>)</span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:3_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E4:3_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">B</font>
;<br/>

<b>thus </b><a NAME="E7:3_1"/>
( ( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E3:3_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ens_1.html#E6:3_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="funct_2.html#T14" title="FUNCT_2:th.14">FUNCT_2:14</a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><a NAME="E3:3"/><i><font color="Green" title="E2">A4</font></i>: 
( ( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> )
 ;<br/>

<a NAME="E4:3"/>
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span>)</span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> : verum </span>}</span>  )
 
<b>by </b><i><a class="txt" href="ens_1.html#E3:3"><i><font color="Green" title="E2">A4</font></i></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E5:3"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ens_1.html#K1" title="ENS_1:func.1">Funcs</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


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