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<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">j</font>, <font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">j</font>;<br/><b>let </b><font color="Maroon">b1'</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">i</font>, <font color="Maroon">b2'</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">i</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:5"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:5"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:5"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2'</font></span>)</span> = 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6">POLYNOM1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b2'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2'</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D6">POLYNOM1:def 6</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:5"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<a NAME="E11:5"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:5"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<i><font color="Green">E51</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2'</font></span>)</span> = 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">i</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2'</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:5"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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