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<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font>, <font color="Maroon" title="c3">I</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">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font><br/> for <font color="Olive" title="b2">h1</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>  st <font color="Olive" title="b1">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Olive" title="b2">h1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> holds <br/><font color="Olive" title="b2">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Olive" title="b1">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a></span><br/><b>let </b><font color="Maroon" title="c4">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</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">h1</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">I</font>  st <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Olive" title="b1">h1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> holds <br/><font color="Olive" title="b1">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</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="c2">H</font>,<font color="Maroon" title="c3">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Maroon" title="c5">h1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> implies <font color="Maroon" title="c5">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> )</span><br/>









<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; <font color="Maroon" title="c5">h1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> )
 ; <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#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a></span><br/>

<a NAME="E2:82"/><b>then </b><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c4">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
;<br/>
<a NAME="E3:82"/><i><font color="Green" title="E56">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</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 <font color="Maroon" title="c3">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<a NAME="E1:82_1"/><i><font color="Green" title="E57">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="c2">H</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="E2:82_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">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 <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></i>;<br/>
<b>hence </b> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</font></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">h1</font>
<b>by </b><i><a class="txt" href="#E1:82_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">I</font>
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:82"/>
<font color="Maroon" title="c5">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</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="#E2:82"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:82"/>
<font color="Maroon" title="c5">h1</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c4">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="#E3:82"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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