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<div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="polyalg1.html#V1" target="_self">strict</a>  <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:46_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:46_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:46_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E5:46_1_2"/><i><font color="Green">E28</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E2:46_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polyalg1.html#E3:46_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E6:46_1_2"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E1:46_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polyalg1.html#E4:46_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E7:46_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E3:46_1_2_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:46_1_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#T15" target="_self">Th15</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E9:46_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E7:46_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E3:46_1_2_2"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:46_1_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#T16" target="_self">Th16</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E11:46_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E7:46_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<i><font color="Green">E33</font></i>: the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>4</sub></font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>
<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>5</sub></font>

;<br/>
<i><font color="Green">E34</font></i>: the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">c<sub>4</sub></font> = 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>
<br/>.= 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">c<sub>5</sub></font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E2:46_1_2_5"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b>the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>7</sub></font>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#T17" target="_self">Th17</a></i>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:46_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E7:46_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polyalg1.html#E9:46_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polyalg1.html#E11:46_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polyalg1.html#E12:46_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polyalg1.html#E13:46_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
