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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">M</font>, <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">N2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">N1</font> is  <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a>  &amp; <font color="Olive" title="b3">N2</font> is  <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a>  &amp; <font color="Olive" title="b1">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Olive" title="b2">N1</font> &amp; <font color="Olive" title="b1">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Olive" title="b3">N2</font> holds <br/><font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">N2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N1</font>, <font color="Maroon" title="c4">N2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</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="c3">N1</font> is  <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a>  &amp; <font color="Maroon" title="c4">N2</font> is  <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c3">N1</font> &amp; <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c4">N2</font> implies <font color="Maroon" title="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">N1</font> is  <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a> 
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c4">N2</font> is  <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a> 
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c3">N1</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c4">N2</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="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N2</font></span><br/>

<a NAME="E5:16"/>
<font color="Maroon" title="c4">N2</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="rusub_5.html#T2" target="_self" title="RUSUB_5:th.2">Th2</a></span>;<br/>
<a NAME="E6:16"/><span class="kw">then </span>
<font color="Maroon" title="c4">N2</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="rusub_5.html#T3" target="_self" title="RUSUB_5:th.3">Th3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E8:16"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c4">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N1</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_5.html#D1" target="_self" title="RUSUB_5:def.1">Def1</a></span>;<br/>
<a NAME="E9:16"/>
<font color="Maroon" title="c3">N1</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="rusub_5.html#T2" target="_self" title="RUSUB_5:th.2">Th2</a></span>;<br/>
<a NAME="E10:16"/><span class="kw">then </span>
<font color="Maroon" title="c3">N1</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c4">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="rusub_5.html#T3" target="_self" title="RUSUB_5:th.3">Th3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E12:16"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N2</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_5.html#D1" target="_self" title="RUSUB_5:def.1">Def1</a></span>;<br/>
<a NAME="E13:16"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="rusub_4.html#D8" title="RUSUB_4:def.8">RUSUB_4:def 8</a></span>;<br/>
<a NAME="E14:16"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">q</font></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N1</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p2</font>, <font color="Maroon" title="c8">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E16:16"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">and </span><br/><a NAME="E17:16"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N1</font>
 <span class="kw">and </span><br/><a NAME="E18:16"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c8">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>
<a NAME="E19:16"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E20:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A10</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="rlvect_1.html#T19" title="RLVECT_1:th.19">RLVECT_1:19</a></span>;<br/>
<font color="Maroon" title="c5">v2</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
;<br/>
<a NAME="E22:16"/><span class="kw">then </span>
<font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="rusub_4.html#D8" title="RUSUB_4:def.8">RUSUB_4:def 8</a></span>;<br/>
<a NAME="E23:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c4">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="rusub_5.html#T13" target="_self" title="RUSUB_5:th.13">Th13</a></span>;<br/>
<a NAME="E24:16"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="rusub_4.html#D8" title="RUSUB_4:def.8">RUSUB_4:def 8</a></span>;<br/>
<a NAME="E25:16"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">q</font></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N2</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p1</font>, <font color="Maroon" title="c10">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E27:16"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">q1</font>
 <span class="kw">and </span><br/><a NAME="E28:16"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c9">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N2</font>
 <span class="kw">and </span><br/><a NAME="E29:16"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c10">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>
<a NAME="E30:16"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E31:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A15</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="rlvect_1.html#T19" title="RLVECT_1:th.19">RLVECT_1:19</a></span>;<br/>
<font color="Maroon" title="c6">v1</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
;<br/>
<a NAME="E33:16"/><span class="kw">then </span>
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E31:16"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="ref" href="rusub_4.html#D8" title="RUSUB_4:def.8">RUSUB_4:def 8</a></span>;<br/>
<a NAME="E34:16"/><span class="kw">then </span>
<font color="Maroon" title="c3">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="rusub_5.html#T13" target="_self" title="RUSUB_5:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:16"/>
<font color="Maroon" title="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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