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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#L1" target="_self">AlgebraStr</a> of <font color="Maroon">L</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#L1" target="_self">AlgebraStr</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> iff <font color="Olive">x</font> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> )
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K8">+</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E3:8_1_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K14">*'</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E4:8_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font><br/> for <font color="Olive">p</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="polynom5.html#K3">*</a> <font color="Olive">p</font>
 <b>and </b><br/><a NAME="E5:8_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:8_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E7:8_1_2"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> iff <font color="Olive">x</font> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> )
 <b>and </b><br/><a NAME="E8:8_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K8">+</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E9:8_1_2"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom3.html#K14">*'</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E10:8_1_2"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font><br/> for <font color="Olive">p</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="polynom5.html#K3">*</a> <font color="Olive">p</font>
 <b>and </b><br/><a NAME="E11:8_1_2"/><i><font color="Green">E88</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E12:8_1_2"/><i><font color="Green">E89</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font>
 ;<br/>

<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:8_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> iff <font color="Maroon">x</font> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:8_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E14:8_1_2"/><b>then </b><i><font color="Green">E91</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="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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/>;<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/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><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">p</font> <a href="polynom3.html#K8">+</a> <font color="Maroon">q</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="E16:8_1_2"/><b>then </b><i><font color="Green">E94</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><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_3"><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/>;<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/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><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">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><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>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><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">a</font>,<font color="Maroon">x</font> = 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>

<br/>.= 
<font color="Maroon">a1</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x1</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">a</font>,<font color="Maroon">x</font>

;<br/></div><b>end;</b></div>
<a NAME="E19:8_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">P1</font> <a href="funct_2.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</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/>
<a NAME="E20:8_1_2"/><i><font color="Green">E98</font></i>: 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:8_1_2"/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E22:8_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="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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