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

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K15">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="group_7.html#K15">*&gt;</a></span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:4_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>and </b><br/><a NAME="E3:4_1_1_1"/><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="topalg_4.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:4_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>and </b><br/><a NAME="E6:4_1_1_1"/><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="topalg_4.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:4_1_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>and </b><br/><a NAME="E9:4_1_1_1"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="topalg_4.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E10:4_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="group_7.html#K15">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_4.html#E2:4_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topalg_4.html#E5:4_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E11:4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="group_7.html#K17">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="topalg_4.html#E8:4_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topalg_4.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="group_7.html#T4">GROUP_7:4</a></i>;<br/><b>then </b><i><font color="Green">E11</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
;<br/><a NAME="E13:4_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K15">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="group_7.html#K15">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_4.html#E2:4_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="topalg_4.html#E5:4_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E14:4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="group_7.html#K17">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i><a class="txt" href="topalg_4.html#E8:4_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topalg_4.html#E2"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="group_7.html#T4">GROUP_7:4</a></i>;<br/><b>then </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
;<br/><b>hence </b><a NAME="E16:4_1_1_1"/>
<span class="p1">(<span class="default"><a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_4.html#E3:4_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="topalg_4.html#E6:4_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="topalg_4.html#E9:4_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="topalg_4.html#E12:4_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="group_7.html#T32">GROUP_7:32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_1_1"/>
 <a href="topalg_4.html#K1" target="_self">Gr2Iso</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> is   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K15">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="group_7.html#K15">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K15">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_7.html#K15">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>


</div>
