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<div class="add">

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon" title="c2">f</font> being  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:51_1_1_1"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 ;<br/><b>take </b><font color="Maroon" title="c3">i</font> =  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f</font>;<br/><b>let </b><font color="Maroon" title="c4">g</font> be  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>assume </b><a NAME="E3:51_1_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_1_1_1_1"/>
( <font color="Maroon" title="c2">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or ( not <font color="Maroon" title="c2">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:51_1_1_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E2:51_1_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="comput_1.html#E3:51_1_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="comput_1.html#E1:51_1_1_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="comput_1.html#D6" target="_self" title="COMPUT_1:def.6">Def6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:51_1_1_1_1_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_1_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:51_1_1_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="comput_1.html#T21" target="_self" title="COMPUT_1:th.21">Th21</a>, <a class="ref" href="comput_1.html#T22" target="_self" title="COMPUT_1:th.22">Th22</a></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="51_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_2"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:51_1_1_1_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c2">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E2:51_1_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="comput_1.html#E3:51_1_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="comput_1.html#E1:51_1_1_1_1_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="comput_1.html#D6" target="_self" title="COMPUT_1:def.6">Def6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:51_1_1_1_1_2"><i><font color="Green" title="E31">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_2_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:51_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:51_1_1_1_1_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="comput_1.html#T21" target="_self" title="COMPUT_1:th.21">Th21</a>, <a class="ref" href="comput_1.html#T22" target="_self" title="COMPUT_1:th.22">Th22</a></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="51_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:51_1_1_1_1_3"/><i><font color="Green" title="E31">A5</font></i>: 
( not <font color="Maroon" title="c2">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c6">X</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_1_1_1_1_3"/><i><font color="Green" title="E32">A6</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:51_1_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="comput_1.html#E3:51_1_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="comput_1.html#D6" target="_self" title="COMPUT_1:def.6">Def6</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>;<br/><a NAME="E5:51_1_1_1_1_3"/><i><font color="Green" title="E33">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</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> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:51_1_1_1_1_3"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><a NAME="E6:51_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font> <b>by </b><i><a class="txt" href="comput_1.html#E3:51_1_1_1_1_3"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/><i><font color="Green" title="E34">A8</font></i>:  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="txt" href="comput_1.html#E5:51_1_1_1_1_3"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
;<br/><b>consider </b><font color="Maroon" title="c9">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font>;<br/><a NAME="E10:51_1_1_1_1_3"/><i><font color="Green" title="E35">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</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> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:51_1_1_1_1_3"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><a NAME="E11:51_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">a</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">X</font> <b>by </b><i><a class="txt" href="comput_1.html#E3:51_1_1_1_1_3"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/><a NAME="E13:51_1_1_1_1_3"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E10:51_1_1_1_1_3"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a></i>;<br/><b>hence </b><a NAME="E14:51_1_1_1_1_3"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:51_1_1_1_1_3"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">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="E2:51_1_1"/>
( ( for <font color="Olive" title="b1">f</font> being <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  holds not <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">F</font> ) implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>


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