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<div class="add">

<b>let </b><font color="Maroon" title="c1">G1</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:83"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> holds  <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
 ;<br/>

<a NAME="E2:83"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</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 <font color="Maroon" title="c1">G1</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/>
<a NAME="E3:83"/><i><font color="Green" title="E38">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</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="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:83_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span> ;<br/>
<a NAME="E4:83_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 
;<br/>
<a NAME="E5:83_1"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<a NAME="E6:83_1"/><b>then </b><i><font color="Green" title="E40">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E7:83_1"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">a</font> <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">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:83_1"/><i><font color="Green" title="E42">A7</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:83_1"/><b>then </b><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="group_7.html#E5:83_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="group_7.html#E6:83_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="group_7.html#E7:83_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E10:83_1"/>
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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 <font color="Olive" title="b1">R</font> )
 
<b>by </b><i><a class="txt" href="group_7.html#E8:83_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="group_7.html#E9:83_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font> = 
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="group_7.html#E1:83"><i><font color="Green" title="E36">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="group_7.html#E7:83_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E13:83_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:83"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:83"/>
<font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:83_2"/><i><font color="Green" title="E39">A9</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c6">n1</font> = <font color="Maroon" title="c4">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> ;<br/>
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">m1</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> = 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m1</font>
<b>by </b><i><a class="txt" href="group_7.html#E1:83"><i><font color="Green" title="E36">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c6">n1</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="group_7.html#E1:83"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="group_7.html#E1:83_2"><i><font color="Green" title="E39">A9</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:83_2"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T97" title="FINSEQ_1:th.97">FINSEQ_1:97</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:83"/>
<font color="Maroon" title="c2">f</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_7.html#E3:83"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="group_6.html#T70" title="GROUP_6:th.70">GROUP_6:70</a></i>;<br/>


</div>
