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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</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"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">a</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">a</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:37_1"/>
<font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><a NAME="E2:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></span>;<br/><a NAME="E3:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E4:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>;<br/><a NAME="E5:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E6:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/><a NAME="E7:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">hence </span><a NAME="E8:37_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></span>; <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="c3">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font></span><br/><span class="kw">assume </span><a NAME="E9:37_1"/>
<font color="Maroon" title="c3">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E10:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></span>;<br/><a NAME="E11:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E12:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E13:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E14:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E15:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/><a NAME="E16:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/><a NAME="E17:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>;<br/><a NAME="E18:37_1"/>
<font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E19:37_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</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="E3:37"/>
( <font color="Maroon" title="c3">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 ; <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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