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<b>let </b><font color="Maroon" title="c1">UA</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font>;<br/><b>let </b><font color="Maroon" title="c3">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">g1</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 ;<br/>
<i><font color="Green" title="E12">A3</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E12"><i><font color="Green" title="E8">Lm2</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">UA</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E12"><i><font color="Green" title="E8">Lm2</font></i></a></i>
;<br/>
<a NAME="E5:14"/>
<font color="Maroon" title="c2">f</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">UA</font>,<font color="Maroon" title="c1">UA</font>
 
<b>by </b><i><a class="ref" href="autalg_1.html#D1" target="_self" title="AUTALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E6:14"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a> <font color="Maroon" title="c1">UA</font>,<font color="Maroon" title="c1">UA</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#D4" title="ALG_1:def.4">ALG_1:def 4</a></i>;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="alg_1.html#D2" title="ALG_1:def.2">ALG_1:def 2</a></i>;<br/>
<a NAME="E8:14"/>
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="autalg_1.html#E3:14"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E10:14"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">UA</font>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T9" target="_self" title="AUTALG_1:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E11:14"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E3:14"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="autalg_1.html#E4:14"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="autalg_1.html#E7:14"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="funct_2.html#T36" title="FUNCT_2:th.36">FUNCT_2:36</a></i>;<br/>


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