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<span class="kw">let </span><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c1">G</font>,<font color="Maroon" title="c2">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#K10" title="GROUP_6:func.10">Image</a> <font color="Olive" title="b1">h</font></span>)</span>,<font color="Maroon" title="c1">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="c1">G</font>,<font color="Maroon" title="c2">H</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>,<font color="Maroon" title="c1">G</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E43">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>,<font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">Imh</font> =  <a href="group_6.html#K10" title="GROUP_6:func.10">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="E3:78"/>
( <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="c1">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#T44" target="_self" onmouseover="rs('group_6/T44')" onmouseout="rh()">Th44</a>, <a class="ref" href="group_6.html#T49" target="_self" onmouseover="rs('group_6/T49')" onmouseout="rh()">Th49</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="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="funct_2.html#T25" onmouseover="rs('funct_2/T25')" onmouseout="rh()">FUNCT_2:25</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Imh</font> holds  <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="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">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="Olive" title="b1">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="Olive" title="b2">b</font></span>)</span></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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c2">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T42" onmouseover="rs('group_2/T42')" onmouseout="rh()">GROUP_2:42</a></span>;<br/><a NAME="E2:78_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>
 ;<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="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:78_1"/><span class="lab"><font color="Green" title="E44">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#T45" target="_self" onmouseover="rs('group_6/T45')" onmouseout="rh()">Th45</a></span>;<br/><a NAME="E5:78_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>
 ;<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="c1">G</font><span class="kw"> such that </span><br/><a NAME="E7:78_1"/><span class="lab"><font color="Green" title="E45">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#T45" target="_self" onmouseover="rs('group_6/T45')" onmouseout="rh()">Th45</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="#E4:78_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E7:78_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="group_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</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#D6" target="_self" onmouseover="rs('group_6/D6')" onmouseout="rh()">Def6</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="#E1:78"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="funct_2.html#T26" onmouseover="rs('funct_2/T26')" onmouseout="rh()">FUNCT_2:26</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="#E1:78"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:78_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="funct_2.html#T26" onmouseover="rs('funct_2/T26')" onmouseout="rh()">FUNCT_2:26</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="#E1:78"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E7:78_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="funct_2.html#T26" onmouseover="rs('funct_2/T26')" onmouseout="rh()">FUNCT_2:26</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:78"/>
<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#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" target="_self" onmouseover="rs('group_6/D6')" onmouseout="rh()">Def6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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