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<div class="add">

<b>let </b><font color="Maroon">G1</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:86"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>
 ;<br/>

<a NAME="E2:86"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:86"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:86_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:86_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span> ;<br/>
<a NAME="E4:86_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>
 
;<br/>
<a NAME="E5:86_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:86_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E7:86_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</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="E8:86_1"/><i><font color="Green">E56</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:86_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="group_7.html#E2:86"><i><font color="Green">E40</font></i></a>, , <a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E10:86_1"/>
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K8" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K8" target="_self">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 
<b>by </b><i><a class="ref" href="group_7.html#D2" target="_self">Def2</a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
<span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E13:86_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:86"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:86_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font>, <font color="Maroon">n1</font> = <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> ;<br/>
<span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">m1</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">n1</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>
<b>by </b><i>, <a class="txt" href="group_7.html#E2:86"><i><font color="Green">E40</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:86_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:86"/>
<font color="Maroon">f</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i>, <a class="ref" href="group_6.html#T70">GROUP_6:70</a></i>;<br/>


</div>
