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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">j</font>;<br/><b>let </b><font color="Maroon" title="c5">b1'</font>, <font color="Maroon" title="c6">b2'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">i</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c5">b1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c6">b2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font> )
 ;<br/>

<a NAME="E2:5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:5"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:5"/><b>then </b><i><font color="Green" title="E9">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<i><font color="Green" title="E10">A5</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c6">b2'</font></span>)</span> = 
<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_2"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="hilbasis.html#E1:5_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="hilbasis.html#E7:5"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="hilbasis.html#E1:5_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c6">b2'</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">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" title="c3">b1</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c6">b2'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="hilbasis.html#E8:5"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<a NAME="E11:5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:5"/><b>then </b><i><font color="Green" title="E11">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<i><font color="Green" title="E12">A8</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c6">b2'</font></span>)</span> = 
<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_4"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">j</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">i</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E12:5"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="hilbasis.html#E1:5_4"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="hilbasis.html#E7:5"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="hilbasis.html#E1:5_4"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c6">b2'</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">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" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1'</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c6">b2'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E12:5"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="hilbasis.html#E13:5"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
