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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:114_1"/>
not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1"><b>suppose </b></a><a NAME="E1:114_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:114_1_1"/><b>then </b>
(  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T14" target="_self">Th14</a>, <a class="ref" href="partfun1.html#T172">PARTFUN1:172</a></i>;<br/><b>hence </b><a NAME="E3:114_1_1"/>
 <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2"><b>suppose </b></a><a NAME="E1:114_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:114_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:114_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:114_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span>
 ;<br/>

<a NAME="E2:114_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T158" target="_self">Th158</a></i>;<br/>
<b>hence </b><a NAME="E3:114_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="funct_2.html#E1:114_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T11" target="_self">Th11</a></i>;<br/>


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

<b>assume </b><a NAME="E2:114_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:114_1_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T121" target="_self">Th121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="funct_2.html#E2:114_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#T121" target="_self">Th121</a></i>;<br/>
<a NAME="E5:114_1_2_1"/>
( <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i><a class="txt" href="funct_2.html#E1:114_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="funct_2.html#E3:114_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T132" target="_self">Th132</a>, <a class="ref" href="partfun1.html#T142">PARTFUN1:142</a></i>;<br/>
<b>hence </b><a NAME="E6:114_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span>
 <b>by </b><i><a class="ref" href="partfun1.html#D7">PARTFUN1:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:114_1_2"/>
 <a href="partfun1.html#K5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3">&lt;:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="partfun1.html#K3">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1" target="_self">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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