<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> <a href="group_9.html#R2" title="GROUP_9:pred.2">are_isomorphic</a>  holds <br/><font color="Olive" title="b2">H</font>,<font color="Olive" title="b1">G</font> <a href="group_9.html#R2" title="GROUP_9:pred.2">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <a href="group_9.html#R2" title="GROUP_9:pred.2">are_isomorphic</a>  implies <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c2">G</font> <a href="group_9.html#R2" title="GROUP_9:pred.2">are_isomorphic</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:47"/>
<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <a href="group_9.html#R2" title="GROUP_9:pred.2">are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c2">G</font> <a href="group_9.html#R2" title="GROUP_9:pred.2">are_isomorphic</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font><span class="kw"> such that </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">Def22</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f9</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/>
<a NAME="E4:47"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">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="c3">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E5:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <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="c3">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<a NAME="E6:47"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <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="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font> holds  <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font>;<br/><span class="kw">thus </span><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font></span>)</span> = 
<font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>
<br/>.= 
<font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">h1</font>, <font color="Maroon" title="c8">h2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h2</font></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">h1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">g2</font> = <font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h2</font>;<br/><a NAME="E1:47_2"/>
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">h2</font></span>)</span> = 
<font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f9</font> = <font color="Maroon" title="c6">f9</font> as   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">f9</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">GROUP_9:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f9</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E12:47"/>
<font color="Maroon" title="c7">f9</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:47"/>
<font color="Maroon" title="c7">f9</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
