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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>assume </b><a NAME="E1:18"/>
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="alg_1.html#R5" target="_self">are_isomorphic</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="alg_1.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:18"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="alg_1.html#R3" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:18"/>
<font color="Maroon">f</font> <a href="alg_1.html#R2" target="_self">is_monomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E41</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font>
<b>by </b><i>, </i>
;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
<br/>.= 
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/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U2</font>,<font color="Maroon">U1</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="alg_1.html#D5" target="_self">ALG_1:def 5</a><br/>

<b>thus </b><a NAME="E10:18"/>
<font color="Maroon">g</font> <a href="alg_1.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 <b>by </b><i>, </i>;<br/>


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