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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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#L1" target="_self">AlgebraStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_1_2"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K8">+</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:8_1_2"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K14">*'</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:8_1_2"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom5.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:8_1_2"/><i><font color="Green">E7</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:8_1_2"/><i><font color="Green">E8</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:8_1_2"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E8:8_1_2"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K8">+</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:8_1_2"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K14">*'</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E10:8_1_2"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom5.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>
 <b>and </b><br/><a NAME="E11:8_1_2"/><i><font color="Green">E13</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E12:8_1_2"/><i><font color="Green">E14</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><i><font color="Green">E15</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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:8_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>4</sub></font> is   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:8_1_2"><i><font color="Green">E3</font></i></a></i>;<br/><b>hence </b><a NAME="E2:8_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="polyalg1.html#E7:8_1_2"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:8_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E13:8_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</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#E13:8_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:8_1_2"><i><font color="Green">E3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E7:8_1_2"><i><font color="Green">E9</font></i></a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom3.html#K8">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E2:8_1_2"><i><font color="Green">E4</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E8:8_1_2"><i><font color="Green">E10</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E16:8_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E14:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><i><font color="Green">E18</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">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E13:8_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</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#E13:8_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:8_1_2"><i><font color="Green">E3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E7:8_1_2"><i><font color="Green">E9</font></i></a></i>;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E3:8_1_2"><i><font color="Green">E5</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E9:8_1_2"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>4</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>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E13:8_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:8_1_2"><i><font color="Green">E3</font></i></a></i>;<br/><b>thus </b>the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="polynom5.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E4:8_1_2"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E10:8_1_2"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E19:8_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E14:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<i><font color="Green">E20</font></i>: the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">c<sub>2</sub></font> = 
 <a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E6:8_1_2"><i><font color="Green">E8</font></i></a></i>
<br/>.= 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E12:8_1_2"><i><font color="Green">E14</font></i></a></i>
;<br/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E5:8_1_2"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E11:8_1_2"><i><font color="Green">E13</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E22:8_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E14:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polyalg1.html#E16:8_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polyalg1.html#E17:8_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="polyalg1.html#E19:8_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polyalg1.html#E20:8_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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