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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algseq_1.html#K2" title="ALGSEQ_1:func.2">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c3">q</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:42"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom3.html#E1:42"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="algseq_1.html#T10" title="ALGSEQ_1:th.10">ALGSEQ_1:10</a></i>;<br/>
<a NAME="E4:42"/>
( <font color="Maroon" title="c5">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c5">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:42_1"/><i><font color="Green" title="E29">A3</font></i>: 
not <font color="Maroon" title="c5">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E2:42_1"/><i><font color="Green" title="E30">A4</font></i>: 
not <font color="Maroon" title="c5">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font>
 ;<br/>

<a NAME="E3:42_1"/>
( <font color="Maroon" title="c5">x1</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c5">x1</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c3">q</font> )
 
<b>by </b><i><a class="txt" href="polynom3.html#E1:42_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="polynom3.html#E2:42_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="polynom3.html#T23" target="_self" title="POLYNOM3:th.23">Th23</a></i>;<br/>
<a NAME="E4:42_1"/><b>then </b>
<font color="Maroon" title="c5">x1</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#T24" target="_self" title="POLYNOM3:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom3.html#E3:42"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="polynom3.html#T23" target="_self" title="POLYNOM3:th.23">Th23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:42"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K2" title="ALGSEQ_1:func.2">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K2" title="ALGSEQ_1:func.2">support</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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