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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">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</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>,<span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E33">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="autalg_1.html#K3" title="AUTALG_1:func.3">UAAutGroup</a> <font color="Maroon" title="c1">UA</font>;<br/>
<b>thus </b><a NAME="E2:50"/>
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:50_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:50"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="autalg_1.html#E52"><i><font color="Green" title="E30">Lm4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D13" title="GROUP_6:def.13">GROUP_6:def 13</a><br/>


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

<b>thus </b><a NAME="E2:50_1"/>
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1_2"/>
<font color="Maroon" title="c2">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D12" title="GROUP_6:def.12">GROUP_6:def 12</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1_2_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <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>  st <font color="Maroon" title="c2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</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="E1:50_1_2_1_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font>
 ;<br/>

<i><font color="Green" title="E35">A3</font></i>: <font color="Maroon" title="c2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> = 
0 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:50"><i><font color="Green" title="E33">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>

;<br/>
<font color="Maroon" title="c2">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> = 
0 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:50"><i><font color="Green" title="E33">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>

;<br/>
<a NAME="E4:50_1_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:50_1_2_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="autalg_1.html#E2:50_1_2_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T134" title="ZFMISC_1:th.134">ZFMISC_1:134</a></i>;<br/>
<b>hence </b><a NAME="E5:50_1_2_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1_2_1"/>
<font color="Maroon" title="c2">h</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="group_6.html#T1" title="GROUP_6:th.1">GROUP_6:1</a></i>;<br/>


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


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


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


</div>
