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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="freealg.html#V3" title="FREEALG:attr.3">free</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font> be    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">B</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c4">h1</font>, <font color="Maroon" title="c5">h2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font>;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">h2</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> $1 holds <br/><font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>;<br/>
<a NAME="E2:60"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:60_1"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#T18" target="_self" title="AOFA_000:th.18">Th18</a></i>;<br/>
<a NAME="E3:60_1"/><b>then </b>
( <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E4:60_1"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:60"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:60"/><i><font color="Green" title="E37">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:60_2"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:60_2"/><i><font color="Green" title="E39">A5</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> )
 ;<br/>

<a NAME="E3:60_2"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:60_2"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">o</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c9">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E5:60_2"/><i><font color="Green" title="E40">A6</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c8">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:60_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="aofa_000.html#T20" target="_self" title="AOFA_000:th.20">Th20</a></i>;<br/>
<a NAME="E6:60_2"/>
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">B</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:60"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></i>;<br/>
<a NAME="E7:60_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T3" title="UNIALG_2:th.3">UNIALG_2:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">o'</font> = <font color="Maroon" title="c8">o</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:60_2"/>
(  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font> &amp;  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pua2mss1.html#K1" title="PUA2MSS1:func.1">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">B</font> )
 
;<br/>
<a NAME="E10:60_2"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c10">o'</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c9">p</font></span>)</span> &amp; <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c10">o'</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h2</font> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c9">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:60"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="aofa_000.html#E5:60_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>now </b></a><div class="add"><a NAME="E1:60_2_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E2:60_2_1"/>
( <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font> &amp; <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">h2</font> )
 ;<br/><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:60_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c6">n</font>
 ;<br/><b>hence </b><a NAME="E4:60_2_1"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:60_2"><i><font color="Green" title="E38">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:60_2"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E2:60_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="aofa_000.html#E5:60_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="aofa_000.html#E10:60_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="aofa_000.html#T1" target="_self" title="AOFA_000:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:60"/><i><font color="Green" title="E38">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="aofa_000.html#E2:60"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="aofa_000.html#E3:60"><i><font color="Green" title="E37">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/><a NAME="E1:60_3"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="ref" href="aofa_000.html#T30" target="_self" title="AOFA_000:th.30">Th30</a></i>;<br/><b>hence </b><a NAME="E2:60_3"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:60"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:60"/>
<font color="Maroon" title="c4">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">h2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
