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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">B</font>, <font color="Olive" title="b2">B1</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">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Olive" title="b2">B1</font> holds <br/><font color="Olive" title="b3">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Olive" title="b2">B1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">B1</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">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font> implies <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:62"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">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#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c2">B</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:62"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c2">B</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">b1</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="E4:62"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">b1</font>
 <span class="kw">and </span><br/><a NAME="E5:62"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c6">x1</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">B</font>
 <span class="kw">and </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">x2</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="E8:62"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> &amp; <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">x2</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="#E5:62"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">x3</font>, <font color="Maroon" title="c10">b2</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="E10:62"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">b2</font>
 <span class="kw">and </span><br/><a NAME="E11:62"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">and </span><br/><a NAME="E12:62"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">x4</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="E14:62"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c9">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x4</font>
 <span class="kw">and </span><br/><a NAME="E15:62"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c11">x4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E36">A7</font></span></a></span>;<br/>
<a NAME="E16:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c11">x4</font></span>)</span> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="mathmorp.html#T3" target="_self" title="MATHMORP:th.3">Th3</a></span>;<br/>
<a NAME="E17:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c9">x3</font></span>)</span> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c6">x1</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="#E8:62"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E18:62"/><span class="kw">then </span>
<font color="Maroon" title="c3">B1</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x3</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">x1</font></span>)</span> <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="ref" href="mathmorp.html#T16" target="_self" title="MATHMORP:th.16">Th16</a></span>;<br/>
<a NAME="E19:62"/><span class="kw">then </span>
<font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">x3</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">B1</font>
 
;<br/>
<a NAME="E20:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">x3</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E37">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:62"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</a> <font color="Maroon" title="c3">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</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/>


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