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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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>, <font color="Olive" title="b3">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B2</font> holds <br/>( <font color="Olive" title="b3">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Olive" title="b2">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Olive" title="b3">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b1">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b2">B2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B1</font>, <font color="Maroon" title="c3">B2</font>, <font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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="c2">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B2</font> implies ( <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B2</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:15"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B2</font>
 ; <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="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:15"/>
<font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</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"> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B2</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B2</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B2</font>
 ; <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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:15_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font>
 <span class="kw">and </span><br/><a NAME="E4:15_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c3">B2</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 ;<br/>
<a NAME="E5:15_1"/>
<font color="Maroon" title="c2">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B2</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="mathmorp.html#T3" target="_self" title="MATHMORP:th.3">Th3</a></span>;<br/>
<a NAME="E6:15_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:15_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c2">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E10">A2</font></span></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>

<span class="kw">let </span><font color="Maroon" title="c5">p</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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c2">B1</font> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:15"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</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"> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B2</font></span><br/>

<a NAME="E4:15"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:15"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E9">A1</font></span></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>
