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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>thus </b><a NAME="E2:48"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="autalg_1.html#K4" target="_self">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="autalg_1.html#K4" target="_self">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:48_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="autalg_1.html#E3:48_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:48_1_1"/>
0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="autalg_1.html#E3:48_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="autalg_1.html#T39" target="_self">Th39</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:48_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E4:48_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:48_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="autalg_1.html#E3:48_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E10:48_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="alg_1.html#R4">is_isomorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="autalg_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:48_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msuhom_1.html#K2">Over</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msuhom_1.html#T20">MSUHOM_1:20</a></i>;<br/>
 <a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">c<sub>6</sub></font> = 
0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="autalg_1.html#E9:48_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msuhom_1.html#D3">MSUHOM_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="autalg_1.html#E3:48_1_1"><i><font color="Green">E34</font></i></a></i>
;<br/>
<a NAME="E13:48_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E7:48_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="autalg_1.html#E11:48_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msuhom_1.html#T9">MSUHOM_1:9</a></i>;<br/>
<b>hence </b><a NAME="E14:48_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="autalg_1.html#K4" target="_self">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="autalg_1.html#E7:48_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="autalg_1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:48_1"/>
 <a href="autalg_1.html#K4" target="_self">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="autalg_1.html#K4" target="_self">MSAAut</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="autalg_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:48_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:48_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="autalg_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_1.html#K13">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E5:48_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></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="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D13">MSUALG_1:def 13</a></i>;<br/>
<a NAME="E6:48_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E7:48_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0;<br/>

<a NAME="E1:48_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msuhom_1.html#K2">Over</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msuhom_1.html#T9">MSUHOM_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="autalg_1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E3:48_1_2_1"/>
 <a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E6:48_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msuhom_1.html#D3">MSUHOM_1:def 3</a></i>;<br/>
<a NAME="E4:48_1_2_1"/><b>then </b>
 <a href="msuhom_1.html#K3">MSAlg</a> <font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msuhom_1.html#K2">Over</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E3:48_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msuhom_1.html#T9">MSUHOM_1:9</a></i>;<br/>
<a NAME="E5:48_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="alg_1.html#R4">is_isomorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="msuhom_1.html#T24">MSUHOM_1:24</a></i>;<br/>
<b>hence </b><a NAME="E6:48_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="autalg_1.html#D1" target="_self">Def1</a></i>;<br/>

<b>hence </b><a NAME="E7:48_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="txt" href="autalg_1.html#E6:48_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:48_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div>
