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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">r</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> <br/> for <font color="Olive" title="b3">e</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> holds <br/> <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b3">e</font>,<font color="Olive" title="b1">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">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"> for <font color="Olive" title="b1">e</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#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> holds <br/> <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b1">e</font>,<font color="Maroon" title="c2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">e</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</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">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> implies  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font></span><br/>

<a NAME="E2:38"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D3" target="_self" title="EUCLID_9:def.3">Def3</a></span>;<br/>
<a NAME="E3:38"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D3" target="_self" title="EUCLID_9:def.3">Def3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </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> ; <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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span> implies <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span>
 ; <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"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c4">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><a NAME="E3:38_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">s</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="euclid_9.html#D3" target="_self" title="EUCLID_9:def.3">Def3</a></span>;<br/><a NAME="E4:38_1"/>
( <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1"/>
<span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="xxreal_1.html#T46" title="XXREAL_1:th.46">XXREAL_1:46</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">hence </span><a NAME="E5:38"/>
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</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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