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<div class="add">

<b>set </b><font color="Maroon" title="c3">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> : <font color="Olive" title="b1">a</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c4">a1</font> being    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">a2</font> = <font color="Maroon" title="c4">a1</font> as    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E3:24_1_1"/>
<font color="Maroon" title="c5">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> : <font color="Olive" title="b1">a</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">D</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> : <font color="Olive" title="b1">a</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<div><i><font color="Green" title="E9">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">a</font> being    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:24_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> )
 ;<br/><b>thus </b><a NAME="E4:24_1_1_1"/>
<font color="Maroon" title="c7">x</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E3:24_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">D'</font> = <font color="Maroon" title="c6">D</font> as    <a href="lmod_7.html#M4" title="LMOD_7:mode.4">LINE_DOMAIN</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="lmod_7.html#D3" target="_self" title="LMOD_7:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">D'</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:24_1_1_2"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D'</font> implies <font color="Maroon" title="c8">x</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E5:24_1_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E2:24_1_1_2"/>
( <font color="Maroon" title="c8">x</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c8">x</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c8">x</font> as    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E3:24_1_1_2_1"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:24_1_1_2_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D'</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:24_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D'</font> iff <font color="Olive" title="b1">x</font> is    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font> )
 ;<br/>


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