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



<b>assume </b><a NAME="E1:18"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="alg_1.html#R5" target="_self">are_isomorphic</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="alg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R3" target="_self">is_epimorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="alg_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:18"/>
<font color="Maroon">c<sub>3</sub></font> <a href="alg_1.html#R2" target="_self">is_monomorphism</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="alg_1.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="alg_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="alg_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then </b><i><font color="Green">E20</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="alg_1.html#E4:18"><i><font color="Green">E18</font></i></a>, <a class="ref" href="alg_1.html#D3" target="_self">Def3</a></i>
;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="alg_1.html#E6:18"><i><font color="Green">E19</font></i></a>, <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">c<sub>1</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="alg_1.html#E7:18"><i><font color="Green">E20</font></i></a>, <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">c<sub>4</sub></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">c<sub>4</sub></font> <a href="alg_1.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="alg_1.html#T11" target="_self">Th11</a></i>;<br/>


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