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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> 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="b2">e1</font>, <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="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="b1">r</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">e1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Olive" title="b3">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</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">e1</font>, <font color="Olive" title="b2">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="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b2">e</font>,<font color="Maroon" title="c2">r</font> holds <br/><font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">e1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Olive" title="b2">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">e1</font>, <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="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">r</font> implies <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c4">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e</font></span>)</span> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">d</font> =  <a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e</font>;<br/>
<a NAME="E1:45"/>
<span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c4">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">e1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c4">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:45"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">r</font> implies <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K3" title="EUCLID:func.3">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e1</font> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <font color="Maroon" title="c4">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K1" title="EUCLID_9:func.1">max_diff_index</a> <font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#T14" target="_self" title="EUCLID_9:th.14">Th14</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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