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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <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="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1"/>
not <!--HUMANRECFAILED-->( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">Y</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="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1"><b>suppose </b></a><a NAME="E1:106_1_1"/>
( <font color="Maroon" title="c2">Y</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="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <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="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/><div class="add"><a NAME="E2:106_1_1"/><b>then </b>
(  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T14" target="_self" title="FUNCT_2:th.14">Th14</a>, <a class="ref" href="partfun1.html#T172" title="PARTFUN1:th.172">PARTFUN1:172</a></i>;<br/><b>hence </b><a NAME="E3:106_1_1"/>
 <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <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="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_2"><b>suppose </b></a><a NAME="E1:106_1_2"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c2">Y</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="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <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="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/><div class="add"><a NAME="E2:106_1_2"/>
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> iff <font color="Olive" title="b1">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> iff <font color="Maroon" title="c3">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font> )</span><br/>

<b>thus </b><a NAME="E1:106_1_2_1"/>
( <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> implies <font color="Maroon" title="c3">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:106_1_2_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>

<a NAME="E2:106_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T158" target="_self" title="FUNCT_2:th.158">Th158</a></i>;<br/>
<b>hence </b><a NAME="E3:106_1_2_1_1"/>
<font color="Maroon" title="c3">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="funct_2.html#E1:106_1_2"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="funct_2.html#T11" target="_self" title="FUNCT_2:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:106_1_2_1"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c3">g</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="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span></span><br/>

<a NAME="E3:106_1_2_1"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c3">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T121" target="_self" title="FUNCT_2:th.121">Th121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">g'</font> = <font color="Maroon" title="c3">g</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="funct_2.html#E2:106_1_2_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="funct_2.html#T121" target="_self" title="FUNCT_2:th.121">Th121</a></i>;<br/>
<a NAME="E5:106_1_2_1"/>
( <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c4">g'</font> &amp; <font color="Maroon" title="c4">g'</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 
<b>by </b><i><a class="txt" href="funct_2.html#E1:106_1_2"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="funct_2.html#E3:106_1_2_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="funct_2.html#T132" target="_self" title="FUNCT_2:th.132">Th132</a>, <a class="ref" href="partfun1.html#T142" title="PARTFUN1:th.142">PARTFUN1:142</a></i>;<br/>
<b>hence </b><a NAME="E6:106_1_2_1"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span>
 <b>by </b><i><a class="ref" href="partfun1.html#D7" title="PARTFUN1:def.7">PARTFUN1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:106_1_2"/>
 <a href="partfun1.html#K5" title="PARTFUN1:func.5">TotFuncs</a> <span class="p1"><a href="partfun1.html#K3" title="PARTFUN1:func.3">&lt;:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="partfun1.html#K3" title="PARTFUN1:func.3">:&gt;</a></span> <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="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
