<?xml version="1.0"?>
<div class="add">

<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">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><br/> for <font color="Olive" title="b2">t</font>, <font color="Olive" title="b3">s</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Olive" title="b1">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font>, <font color="Maroon" title="c4">s</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span></span><br/>







<span class="kw">thus </span><a NAME="E1:68"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">proof </span></a><div class="add">

<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</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="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</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="E3:68_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>
<a NAME="E4:68_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">z</font></span>)</span> &amp; <font color="Maroon" title="c4">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:68_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</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">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="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:68"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">t</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</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="E4:68"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">and </span><br/><a NAME="E5:68"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">s</font> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">z1</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="E7:68"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">z1</font>
 <span class="kw">and </span><br/><a NAME="E8:68"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<a NAME="E9:68"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:68"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="mathmorp.html#K7" title="MATHMORP:func.7">(.)</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E35">A5</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>
