<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">FV</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">T</font>,<font color="Olive">TT</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">T</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K17" target="_self">TOL</a> <font color="Maroon">X</font>, <font color="Olive">TT</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K17" target="_self">TOL</a> <font color="Maroon">X</font>, <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K20" target="_self">FuncsT</a> <font color="Maroon">X</font> : ( ( <font color="Olive">TT</font> <a href="coh_sp.html#K18" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Olive">T</font> <a href="coh_sp.html#K18" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Olive">T</font> <a href="coh_sp.html#K18" target="_self">`2</a> ,<font color="Olive">TT</font> <a href="coh_sp.html#K18" target="_self">`2</a>  &amp; ( 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> <font color="Olive">T</font> <a href="coh_sp.html#K19" target="_self">`1</a>  holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">TT</font> <a href="coh_sp.html#K19" target="_self">`1</a>  ) ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>set </b><font color="Maroon">a</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span>;<br/><b>set </b><font color="Maroon">f</font> =  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>;<br/><b>take </b><font color="Maroon">m</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">a</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K17" target="_self">TOL</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E3:64_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">a</font> <a href="coh_sp.html#K18" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a</font> <a href="coh_sp.html#K18" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">f</font> =  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="toler_1.html#NK1" target="_self">Total</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K20" target="_self">FuncsT</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<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> <font color="Maroon">a</font> <a href="coh_sp.html#K19" target="_self">`1</a> 
 ;<br/><a NAME="E2:64_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="coh_sp.html#K18" target="_self">`2</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="coh_sp.html#K18" target="_self">`2</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:64_1_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1_1_1"/>
<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> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="coh_sp.html#K19" target="_self">`1</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:64_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">T</font>,<font color="Olive">TT</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">T</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K17" target="_self">TOL</a> <font color="Maroon">X</font>, <font color="Olive">TT</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K17" target="_self">TOL</a> <font color="Maroon">X</font>, <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K20" target="_self">FuncsT</a> <font color="Maroon">X</font> : ( ( <font color="Olive">TT</font> <a href="coh_sp.html#K18" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Olive">T</font> <a href="coh_sp.html#K18" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Olive">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Olive">T</font> <a href="coh_sp.html#K18" target="_self">`2</a> ,<font color="Olive">TT</font> <a href="coh_sp.html#K18" target="_self">`2</a>  &amp; ( 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> <font color="Olive">T</font> <a href="coh_sp.html#K19" target="_self">`1</a>  holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">TT</font> <a href="coh_sp.html#K19" target="_self">`1</a>  ) ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:64_1_1"/>
not  <a href="coh_sp.html#K21" target="_self">MapsT</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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