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<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U2</font>,<font color="Maroon">U1</font>;<br/>







<b>assume </b><a NAME="E1:14"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">h</font> <a href="alg_1.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  )
 ;<br/>

<a NAME="E2:14"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">h</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> &amp; <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:14"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:14"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E6:14"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font>
 ;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font>;<br/><b>assume </b><a NAME="E2:14_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">U2</font>;<br/><b>assume </b><a NAME="E3:14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o2</font>
 ;<br/><a NAME="E4:14_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/><a NAME="E5:14_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E7:14_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E8:14_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> holds <br/> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">U1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">a</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>

<a NAME="E2:14_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:14_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:14_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">U1</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E7:14_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">a</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">U1</font><b> such that </b><br/><a NAME="E10:14_1"/><i><font color="Green">E71</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><i><font color="Green">E72</font></i>: <font color="Maroon">h1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> = 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/><a NAME="E12:14_1"/><i><font color="Green">E73</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:14_1_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, </i>
;<br/><br/></div><b>end;</b></div><a NAME="E14:14_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/><b>then </b><i><font color="Green">E76</font></i>: <font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i/>
;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E17:14_1"/>
( <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> )
 <b>by </b><i>, , , , , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><a NAME="E18:14_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
 <b>by </b><i>, , </i>;<br/><a NAME="E19:14_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/><a NAME="E20:14_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/><a NAME="E21:14_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="alg_1.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E22:14_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E23:14_1"/><b>then </b>
<font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:14"/>
<font color="Maroon">h1</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 <b>by </b><i>, </i>;<br/>


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