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<span class="kw">let </span><font color="Maroon" title="c1">H</font>, <font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">H</font>  st <font color="Olive" title="b1">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/><font color="Olive" title="b1">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Olive" title="b1">h</font></span>)</span>,<font color="Maroon" title="c2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">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="c3">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>,<font color="Maroon" title="c2">G</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c3">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>,<font color="Maroon" title="c2">G</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">Imh</font> =  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">h</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<a NAME="E3:84"/>
( <font color="Maroon" title="c3">h</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">Imh</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">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="c4">Imh</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T53" target="_self" title="GROUP_6:th.53">Th53</a>, <a class="ref" href="group_6.html#T58" target="_self" title="GROUP_6:th.58">Th58</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">h9</font> = <font color="Maroon" title="c3">h</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="c4">Imh</font>,<font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Imh</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="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</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="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/><a NAME="E2:84_1"/>
<font color="Maroon" title="c8">a9</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">Imh</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E4:84_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></span>;<br/><a NAME="E5:84_1"/>
<font color="Maroon" title="c9">b9</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">Imh</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E7:84_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> = 
<font color="Maroon" title="c5">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E7:84_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></span>
<br/>.= 
<font color="Maroon" title="c5">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c10">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E4:84_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</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="c5">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E7:84_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</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>
<span class="kw">hence </span><a NAME="E6:84"/>
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>,<font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</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/>


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