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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="seqm_3.html#V6">natural-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:91_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:91_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:91_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E4:91_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><b>hence </b><a NAME="E5:91_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D5" target="_self">Def5</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:91_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>assume </b><a NAME="E3:91_1"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E4:91_1"/><b>then </b>
( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:91_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E6:91_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E7:91_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E2:91_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="polynom1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:91"/>
 <a href="polynom1.html#K11" target="_self">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" target="_self">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" target="_self">Def7</a></i>;<br/>


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