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

<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font>, <font color="Maroon" title="c3">H1</font>, <font color="Maroon" title="c4">H2</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">H1</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">H2</font>  st <font color="Olive" title="b1">f</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  &amp; <font color="Olive" title="b2">g</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  holds <br/> <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/><b>let </b><font color="Maroon" title="c5">h1</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">H1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">H2</font>  st <font color="Maroon" title="c5">h1</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  holds <br/> <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Olive" title="b1">g</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/><b>let </b><font color="Maroon" title="c6">h2</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">H2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">h1</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  &amp; <font color="Maroon" title="c6">h2</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  implies  <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c5">h1</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> </span><br/>

<b>set </b><font color="Maroon" title="c7">h</font> =  <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font>;<br/>
<a NAME="E3:7"/>
( <font color="Maroon" title="c5">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c6">h2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="topalg_4.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="group_6.html#T70" title="GROUP_6:th.70">GROUP_6:70</a></i>;<br/>
<a NAME="E4:7"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
 <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</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="topalg_4.html#T3" target="_self" title="TOPALG_4:th.3">Th3</a></i>;<br/>
<a NAME="E5:7"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h1</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">H1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</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="c4">H2</font> )
 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="topalg_4.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="group_6.html#T70" title="GROUP_6:th.70">GROUP_6:70</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
( <font color="Maroon" title="c5">h1</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c6">h2</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E7:7"/><b>then </b>
 <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="topalg_4.html#T4" target="_self" title="TOPALG_4:th.4">Th4</a></i>;<br/>
<a NAME="E8:7"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font></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 <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#K15" title="GROUP_7:func.15">&lt;*</a><span class="default"><font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c4">H2</font></span><a href="group_7.html#K15" title="GROUP_7:func.15">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:7"/>
 <a href="topalg_4.html#K2" title="TOPALG_4:func.2">Gr2Iso</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c6">h2</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_4.html#E4:7"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="group_6.html#T70" title="GROUP_6:th.70">GROUP_6:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
