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<span class="kw">let </span><font color="Maroon" title="c1">C1</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_1.html#L1" title="ALTCAT_1:struct.1">AltGraph</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">C2</font>, <font color="Olive" title="b2">C3</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_2.html#V1" title="ALTCAT_2:attr.1">reflexive</a>   <a href="altcat_1.html#L1" title="ALTCAT_1:struct.1">AltGraph</a> <br/> for <font color="Olive" title="b3">F</font> being  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">C1</font>,<font color="Olive" title="b1">C2</font><br/> for <font color="Olive" title="b4">G</font> being   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Olive" title="b1">C2</font>,<font color="Olive" title="b2">C3</font>  st <font color="Olive" title="b3">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  &amp; <font color="Olive" title="b4">G</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  holds <br/><font color="Olive" title="b4">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Olive" title="b3">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c3">C3</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_2.html#V1" title="ALTCAT_2:attr.1">reflexive</a>   <a href="altcat_1.html#L1" title="ALTCAT_1:struct.1">AltGraph</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">F</font> being  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font>  st <font color="Olive" title="b1">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  &amp; <font color="Olive" title="b2">G</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  holds <br/><font color="Olive" title="b2">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Olive" title="b1">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>; <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">G</font> being   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font>  st <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  holds <br/><font color="Olive" title="b1">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be    <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</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="c4">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  implies <font color="Maroon" title="c5">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> 
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</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">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">MMG</font> = the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font>;<br/>
<a NAME="E3:11"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">MMF</font> = the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">CC2</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">CC1</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">MMGF</font> = the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">F</font></span>)</span> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">OMF</font> = the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c4">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E6:11"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c10">MMGF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">OMF</font></span>)</span> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D37" title="FUNCTOR0:def.37">FUNCTOR0:def 37</a></span>;<br/>
<a NAME="E7:11"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a></span>;<br/>
<a NAME="E8:11"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c10">MMGF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</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="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c10">MMGF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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="c10">MMGF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<a NAME="E2:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">OMF</font></span>)</span> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E12">A7</font></span>: <font color="Maroon" title="c10">MMGF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">OMF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">OMF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
;<br/>
<a NAME="E4:11_1"/>
<font color="Maroon" title="c11">OMF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E5:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A8</font></span>: 
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">OMF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></span>;<br/>
<a NAME="E6:11_1"/>
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:11_1"/>
<font color="Maroon" title="c10">MMGF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E13">A8</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:11"/>
the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c4">F</font></span>)</span> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a> <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>
