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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b3">T</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">W</font><br/> for <font color="Olive" title="b4">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">V</font><br/> for <font color="Olive" title="b5">l</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b3">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b4">X</font><br/> for <font color="Olive" title="b6">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">V</font>  st <font color="Olive" title="b6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">X</font> &amp; <font color="Olive" title="b3">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b4">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Olive" title="b5">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">W</font>, <font color="Maroon" title="c3">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c2">W</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b3">l</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b2">X</font><br/> for <font color="Olive" title="b4">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b1">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Olive" title="b3">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c2">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b2">l</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Olive" title="b2">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font> being   <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c5">X</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">l</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span> )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">X</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span></span><br/>

<a NAME="E3:57"/><i><font color="Green" title="E45">A3</font></i>: 
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V</font> <a href="ranknull.html#K4" title="RANKNULL:func.4">\</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E1:57"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:57"/>
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V</font> <a href="ranknull.html#K4" title="RANKNULL:func.4">\</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E2:57"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="ranknull.html#D6" target="_self" title="RANKNULL:def.6">Def6</a></i>;<br/>
<a NAME="E5:57"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E3:57"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/>
<a NAME="E6:57"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="ranknull.html#T7" target="_self" title="RANKNULL:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:57"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K9" title="RANKNULL:func.9">#</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="ranknull.html#E5:57"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
