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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">U3</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">h1</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">h2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U2</font>,<font color="Maroon">U3</font>;<br/>









<b>set </b><font color="Maroon">s1</font> =  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font>;<br/>
<b>set </b><font color="Maroon">s2</font> =  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font>;<br/>
<b>set </b><font color="Maroon">s3</font> =  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U3</font>;<br/>
<b>assume </b><a NAME="E1:11"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">h1</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">h2</font> <a href="alg_1.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font> )
 ;<br/>

<a NAME="E2:11"/><b>then </b>
( <font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a>  &amp; <font color="Maroon">U2</font>,<font color="Maroon">U3</font> <a href="unialg_2.html#R1">are_similar</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green">E34</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> &amp;  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U3</font> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:11"/>
 <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">U3</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a>,<a class="ref" href="alg_1.html#D1" target="_self">ALG_1:def 1</a><br/>

<a NAME="E5:11"/><i><font color="Green">E35</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="E6:11"/><i><font color="Green">E41</font></i>: 
(  <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> &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">U1</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">U1</font></span>)</span></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> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<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="E7:11"/><i><font color="Green">E46</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">U1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">o2</font> = 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> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font> <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<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>let </b><font color="Maroon">o3</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U3</font>;<br/>



<b>assume </b><a NAME="E9:11"/><i><font color="Green">E47</font></i>: 
( <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> &amp; <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">U1</font>;<br/>

<b>assume </b><a NAME="E10:11"/><i><font color="Green">E48</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">o1</font>
 ;<br/>

<a NAME="E11:11"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">o1</font> <a href="tarski.html#R1">c=</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_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:11"/><i><font color="Green">E50</font></i>: 
( <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="E13:11"/><i><font color="Green">E61</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</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> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</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="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E14:11"/><b>then </b>
<font color="Maroon">a</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>, <a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <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> such that </b><br/><a NAME="E16:11"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font> as    <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>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E18:11"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E19:11"/><b>then </b>
<font color="Maroon">b</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> 
 
;<br/>
<a NAME="E20:11"/><b>then </b>
<font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<a NAME="E21:11"/><b>then </b>
( <font color="Maroon">h1</font> <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">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</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">a</font></span>)</span> &amp; <font color="Maroon">h2</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> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseqop.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , , , , , </i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</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">a</font></span>)</span> = 
<font color="Maroon">o3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseqop.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">o3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
;<br/><br/>


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