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

<b>let </b><font color="Maroon">w1</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font>, <font color="Maroon">w2</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14_1_2"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">b</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  ex <font color="Olive">b1</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> ex <font color="Olive">Q1</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font> &amp; <font color="Olive">b</font> <a href="pboole.html#R6">=</a> <font color="Olive">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> &amp; <font color="Maroon">w1</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font> )
 <b>and </b><br/><a NAME="E2:14_1_2"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">b</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  ex <font color="Olive">b1</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> ex <font color="Olive">Q1</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font> &amp; <font color="Olive">b</font> <a href="pboole.html#R6">=</a> <font color="Olive">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> &amp; <font color="Maroon">w2</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font> )
 ;<br/>

<a NAME="E3:14_1_2"/>
for <font color="Olive">c</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> holds  <font color="Maroon">w1</font> <a href="polynom1.html#K15">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="polynom1.html#K15">.</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">c2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:14_1_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c1</font>
 <b>and </b><br/><a NAME="E3:14_1_2_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">c</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c2</font>
 <b>and </b><br/><a NAME="E4:14_1_2_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">w1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">d2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">S1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:14_1_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">d1</font>
 <b>and </b><br/><a NAME="E7:14_1_2_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">c</font> <a href="pboole.html#R6">=</a> <font color="Maroon">d1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">d2</font>
 <b>and </b><br/><a NAME="E8:14_1_2_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">w2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">d2</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:14_1_2_1"/>
<font color="Maroon">c2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">d2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:14_1_2_1"/>
<font color="Maroon">w1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1_2"/>
<font color="Maroon">w1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
