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<div class="add">

<b>let </b><font color="Maroon" title="c5">w1</font>, <font color="Maroon" title="c6">w2</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14_1_2"/><i><font color="Green" title="E9">A41</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">b1</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> ex <font color="Olive" title="b4">Q1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b2</font> &amp; <font color="Maroon" title="c5">w1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font> )
 <b>and </b><br/><a NAME="E2:14_1_2"/><i><font color="Green" title="E10">A42</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">b1</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font> ex <font color="Olive" title="b4">Q1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b2</font> &amp; <font color="Maroon" title="c6">w2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font> )
 ;<br/>

<a NAME="E3:14_1_2"/>
for <font color="Olive" title="b1">c</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <font color="Maroon" title="c5">w1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">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" title="c7">c</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c8">c1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c9">c2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c10">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E2:14_1_2_1"/><i><font color="Green" title="E11">A43</font></i>: 
<font color="Maroon" title="c10">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c8">c1</font>
 <b>and </b><br/><a NAME="E3:14_1_2_1"/><i><font color="Green" title="E12">A44</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c9">c2</font>
 <b>and </b><br/><a NAME="E4:14_1_2_1"/><i><font color="Green" title="E13">A45</font></i>: 
<font color="Maroon" title="c5">w1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Q1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c9">c2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:14_1_2"><i><font color="Green" title="E9">A41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">d1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c12">d2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c13">S1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E6:14_1_2_1"/><i><font color="Green" title="E14">A46</font></i>: 
<font color="Maroon" title="c13">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c11">d1</font>
 <b>and </b><br/><a NAME="E7:14_1_2_1"/><i><font color="Green" title="E15">A47</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">d1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c12">d2</font>
 <b>and </b><br/><a NAME="E8:14_1_2_1"/><i><font color="Green" title="E16">A48</font></i>: 
<font color="Maroon" title="c6">w2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c12">d2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:14_1_2"><i><font color="Green" title="E10">A42</font></i></a></i>;<br/>
<a NAME="E9:14_1_2_1"/>
<font color="Maroon" title="c9">c2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c12">d2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:14_1_2_1"><i><font color="Green" title="E12">A44</font></i></a>, <a class="txt" href="polynom6.html#E7:14_1_2_1"><i><font color="Green" title="E15">A47</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E10:14_1_2_1"/>
<font color="Maroon" title="c5">w1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:14_1_2_1"><i><font color="Green" title="E11">A43</font></i></a>, <a class="txt" href="polynom6.html#E3:14_1_2_1"><i><font color="Green" title="E12">A44</font></i></a>, <a class="txt" href="polynom6.html#E4:14_1_2_1"><i><font color="Green" title="E13">A45</font></i></a>, <a class="txt" href="polynom6.html#E6:14_1_2_1"><i><font color="Green" title="E14">A46</font></i></a>, <a class="txt" href="polynom6.html#E7:14_1_2_1"><i><font color="Green" title="E15">A47</font></i></a>, <a class="txt" href="polynom6.html#E8:14_1_2_1"><i><font color="Green" title="E16">A48</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1_2"/>
<font color="Maroon" title="c5">w1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">w2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
