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<b>let </b><font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">P2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c2">B</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_2"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</font>
 <b>and </b><br/><a NAME="E2:46_1_2"/><i><font color="Green" title="E25">A8</font></i>: 
for <font color="Olive" title="b1">C</font> being   <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font> holds <br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font>
 <b>and </b><br/><a NAME="E3:46_1_2"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font>
 <b>and </b><br/><a NAME="E4:46_1_2"/><i><font color="Green" title="E27">A10</font></i>: 
for <font color="Olive" title="b1">C</font> being   <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font> holds <br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font>
 ;<br/>

<a NAME="E5:46_1_2"/><i><font color="Green" title="E28">A11</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E2:46_1_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="polyalg1.html#E3:46_1_2"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>
<a NAME="E6:46_1_2"/><i><font color="Green" title="E29">A12</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E1:46_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="polyalg1.html#E4:46_1_2"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E7:46_1_2"/><b>then </b><i><font color="Green" title="E30">A13</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</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="c5">P2</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P1</font>;<br/><b>let </b><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">P2</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">P2</font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P1</font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E3:46_1_2_1"/><i><font color="Green" title="E31">A14</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">x'</font> = <font color="Maroon" title="c8">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:46_1_2_1"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">P1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c9">y1</font>

<br/>.= 
<font color="Maroon" title="c10">x'</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c11">y'</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T15" target="_self" title="POLYALG1:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T15" target="_self" title="POLYALG1:th.15">Th15</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c5">P2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>

;<br/></div><b>end;</b></div>
<a NAME="E9:46_1_2"/><b>then </b><i><font color="Green" title="E31">A15</font></i>: 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">P1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c5">P2</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E7:46_1_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">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" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P1</font>;<br/><b>let </b><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">P2</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">P2</font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P1</font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E3:46_1_2_2"/><i><font color="Green" title="E32">A16</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">x'</font> = <font color="Maroon" title="c8">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="polyalg1.html#E3:46_1_2_2"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c4">P1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c6">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">y1</font>

<br/>.= 
<font color="Maroon" title="c10">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">y'</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T16" target="_self" title="POLYALG1:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c8">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T16" target="_self" title="POLYALG1:th.16">Th16</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c5">P2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>

;<br/></div><b>end;</b></div>
<a NAME="E11:46_1_2"/><b>then </b><i><font color="Green" title="E32">A17</font></i>: 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c4">P1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c5">P2</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E7:46_1_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>
<i><font color="Green" title="E33">A18</font></i>: the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c4">P1</font> = 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">P1</font>

<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">B</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">P2</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c5">P2</font>

;<br/>
<i><font color="Green" title="E34">A19</font></i>: the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c4">P1</font> = 
 <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c4">P1</font>

<br/>.= 
 <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c2">B</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
 <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c5">P2</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c5">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" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">P1</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">P2</font> <b>by </b><i><a class="txt" href="polyalg1.html#E5:46_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="polyalg1.html#E6:46_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E2:46_1_2_5"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c8">y1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b>the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c4">P1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c6">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">y</font>

<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">y'</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T17" target="_self" title="POLYALG1:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">y1</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T17" target="_self" title="POLYALG1:th.17">Th17</a></i>
<br/>.= 
the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c5">P2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:46_1_2"/>
<font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E7:46_1_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="polyalg1.html#E9:46_1_2"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="polyalg1.html#E11:46_1_2"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="polyalg1.html#E12:46_1_2"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="polyalg1.html#E13:46_1_2"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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