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<b>let </b><font color="Maroon" title="c1">G1</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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font>,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#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>  st ( for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> ) holds <br/><font color="Olive" title="b1">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<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#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font>,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#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>; <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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> holds  <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> ) implies <font color="Maroon" title="c2">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<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#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> holds  <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<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#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span></span><br/><b>thus </b><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span> = 
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="group_7.html#E1:82"><i><font color="Green" title="E35">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="group_7.html#T31" target="_self" title="GROUP_7:th.31">Th31</a></i>
<br/>.= 
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_7.html#E1:82"><i><font color="Green" title="E35">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_7.html#E1:82"><i><font color="Green" title="E35">A1</font></i></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><b>end;</b></div>
<b>hence </b><a NAME="E3:82"/>
<font color="Maroon" title="c2">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<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#K8" title="GROUP_7:func.8">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K8" title="GROUP_7:func.8">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</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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