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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> holds   <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Olive" title="b1">b1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c2">b1</font></span><br/>



<span class="kw">thus </span><a NAME="E1:99"/>
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c2">b1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">b2</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c2">b1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_1"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">b2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c2">b1</font></span><br/>

<span class="kw">assume </span><a NAME="E2:99_1"/>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c2">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:99_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" target="_self" title="PRE_POLY:def.7">Def7</a></span>;<br/>
<a NAME="E4:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></span>;<br/>
<a NAME="E5:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c3">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" target="_self" title="PRE_POLY:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:99_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="pre_poly.html#D7" target="_self" title="PRE_POLY:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


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