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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>assume </b><a NAME="E1:26"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">h</font> <a href="alg_1.html#R2">is_monomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</font>;<br/>
<a NAME="E2:26"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">h</font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="alg_1.html#D2">ALG_1:def 2</a></i>;<br/>
<a NAME="E3:26"/><b>then </b>
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i><a class="ref" href="alg_1.html#D1">ALG_1:def 1</a></i>;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:26"/>
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span> <a href="msuhom_1.html#K2" target="_self">Over</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a><br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a><br/><b>let </b><font color="Maroon">h'</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E6:26"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</font></span>)</span>
 <b>and </b><br/><a NAME="E7:26"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">g</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">U1</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">U1</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_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E9:26"/><i><font color="Green">E53</font></i>: 
( 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">U1</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">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="msualg_1.html#K6">signature</a> <font color="Maroon">U1</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">U1</font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</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">U1</font></span>)</span> <a href="hidden.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">U1</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D13">MSUALG_1:def 13</a></i>;<br/>
<a NAME="E10:26"/>
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">h</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:26"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E12:26"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h</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>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E13:26"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:26"/>
<font color="Maroon">h'</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="alg_1.html#D2">ALG_1:def 2</a></i>;<br/>


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