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<b>let </b><font color="Maroon">P1</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">B</font>, <font color="Maroon">P2</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">B</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font>
 <b>and </b><br/><a NAME="E2:46_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>  st <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font>
 <b>and </b><br/><a NAME="E3:46_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font>
 <b>and </b><br/><a NAME="E4:46_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">C</font> being   <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>  st <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font> holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C</font>
 ;<br/>

<a NAME="E5:46_1_2"/><i><font color="Green">E58</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:46_1_2"/><i><font color="Green">E59</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:46_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, <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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font> <b>by </b><i>, , <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">E61</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, <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">P1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y1</font>

<br/>.= 
<font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>

;<br/></div><b>end;</b></div>
<a NAME="E9:46_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">P1</font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, <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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font> <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font> <b>by </b><i>, , <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">E65</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, <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">P1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y1</font>

<br/>.= 
<font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>

;<br/></div><b>end;</b></div>
<a NAME="E11:46_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">P1</font> <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<i><font color="Green">E67</font></i>: the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">P1</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P1</font>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P2</font>
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">P2</font>

;<br/>
<i><font color="Green">E68</font></i>: the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">P1</font> = 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">P1</font>

<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font>
<b>by </b><i/>
<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">P2</font>
<b>by </b><i/>
<br/>.= 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">P2</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font> <b>by </b><i>, , <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">P2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</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">P1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font>

<br/>.= 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y1</font>
<b>by </b><i/>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:46_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , , , <a class="ref" href="polyalg1.html#T1" target="_self">Th1</a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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