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<div class="add">

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">f</font> being  <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E1:53_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/><b>take </b><font color="Maroon">i</font> =  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><a NAME="E2:53_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1"/>
( <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> or ( not <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:53_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1_1_1_1"/>
( <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> or  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1_1_1_1"/>
<font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:53_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1_2_1_1"/>
( <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> or  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2_1_1_1"/>
<font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_3"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">X</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_1_1_3"/><i><font color="Green">E54</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>;<br/><a NAME="E5:53_1_1_1_1_3"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E6:53_1_1_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/><i><font color="Green">E57</font></i>:  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>;<br/><a NAME="E10:53_1_1_1_1_3"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E11:53_1_1_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E13:53_1_1_1_1_3"/>
 <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E14:53_1_1_1_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:53_1_1"/>
( ( for <font color="Olive">f</font> being <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>  holds not <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> ) implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


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