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

<b>set </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> ;<br/>
<a NAME="E1:55_1_1"/>
( not <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span>  is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span>  is <a href="fraenkel.html#V1">functional</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/>
<a NAME="E2:55_1_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<a NAME="E3:55_1_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:55_1_1_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E5:55_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> 
 ;<br/>

<a NAME="E6:55_1_1_1"/><b>then </b>
 ex <font color="Olive">g</font> being   <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> )
 
;<br/>
<b>hence </b><a NAME="E7:55_1_1_1"/>
<font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H</font>;<br/><a NAME="E1:55_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:55_1_1_2"/><b>then </b>
 ex <font color="Olive">g</font> being   <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> )
 ;<br/><b>hence </b><a NAME="E3:55_1_1_2"/>
<font color="Maroon">f</font> is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">X</font> : <font color="Olive">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span>  is   <a href="rfunct_3.html#NM2">PFUNC_DOMAIN</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rfunct_3.html#D3">RFUNCT_3:def 3</a></i>;<br/>


</div>
