<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c4">a1</font> =  the    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">a2</font> =  the    <a href="lmod_7.html#M3" title="LMOD_7:mode.3">LINE</a> of <font color="Maroon" title="c2">V</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/>
<a NAME="E2:24_1_1"/>
<font color="Maroon" title="c5">a2</font> <a href="tarski.html#R2" title="TARSKI: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/>
<span class="kw">then </span><span class="kw">reconsider </span><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#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<div><span class="lab"><font color="Green" title="E9">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">D</font> holds <br/><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></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">D</font> implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><a NAME="E2:24_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <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> )
 ;<br/><span class="kw">hence </span><a NAME="E3:24_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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">D9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#D3" target="_self" onmouseover="rs('lmod_7/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">D9</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">D9</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> )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">D9</font> implies <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> ) &amp; ( <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> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D9</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D9</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> ) &amp; ( <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">D9</font> ) )</span><br/><span class="kw">thus </span><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">D9</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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">D9</font> )</span><br/><span class="kw">thus </span><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">D9</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_1_2_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D9</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/>
<a NAME="E3:24_1_1_2_1"/>
<font color="Maroon" title="c9">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_2_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><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">D9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:24_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">D9</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> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
