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<span class="kw">let </span><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">W</font> be   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>   <a href="zmodul03.html#V1" title="ZMODUL03:attr.1">finite-rank</a>  <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</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">  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="c1">V</font>,<font color="Maroon" title="c2">W</font>  st <font color="Olive" title="b1">T</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">W</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="c3">T</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:48"/>
<font color="Maroon" title="c3">T</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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">  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font></span><br/>

<a NAME="E2:48"/><span class="kw">then </span>
 <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T25" onmouseover="rs('zmodul05/T25')" onmouseout="rh()">ZMODUL05:25</a></span>;<br/>
<a NAME="E3:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="zmodul07.html#K5" title="ZMODUL07:func.5">nullity</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul05.html#T26" onmouseover="rs('zmodul05/T26')" onmouseout="rh()">ZMODUL05:26</a></span>;<br/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> = 
<span class="p1">(<span class="default"><a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K5" title="ZMODUL07:func.5">nullity</a> <font color="Maroon" title="c3">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T24" target="_self" onmouseover="rs('zmodul07/T24')" onmouseout="rh()">Th44</a></span>
<br/>.= 
 <a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:48"/>
 <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zmodul07.html#K4" title="ZMODUL07:func.4">rank</a> <font color="Maroon" title="c3">T</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/>


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