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<b>let </b><font color="Maroon" title="c1">K</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">V1</font>, <font color="Olive" title="b2">V2</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V1</font><br/> for <font color="Olive" title="b4">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V2</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">V2</font>  st  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">V2</font> holds <br/>( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b5">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Olive" title="b5">f</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</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">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</font><br/> for <font color="Olive" title="b2">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V2</font> holds <br/>( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b3">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c4">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</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">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V2</font> holds <br/>( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b2">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Olive" title="b1">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c5">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</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">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V2</font> holds <br/>( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b1">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V2</font> implies ( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> ) )</span><br/>











<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E39">A1</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> iff  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )</span><br/>

<b>set </b><font color="Maroon" title="c7">A</font> =  <a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font>;<br/>
<a NAME="E2:69"/><i><font color="Green" title="E40">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font> &amp;  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a>, <a class="ref" href="matrlin2.html#T48" target="_self" title="MATRLIN2:th.48">Th48</a></i>;<br/>
<a NAME="E3:69"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
(  <a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c6">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrlin2.html#E1:69"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="matrlin2.html#D2" target="_self" title="MATRLIN2:def.2">Def2</a>, <a class="ref" href="ranknull.html#T44" title="RANKNULL:th.44">RANKNULL:44</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> implies  not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:69_1"/>
not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span><br/><a NAME="E2:69_1"/><b>then </b>
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font>
 <b>by </b><i><a class="txt" href="matrlin2.html#E3:69"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="matrlin2.html#T42" target="_self" title="MATRLIN2:th.42">Th42</a></i>;<br/><b>hence </b><a NAME="E3:69_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="matrlin2.html#E2:69"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin2.html#E3:69"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="matrix13.html#T83" title="MATRIX13:th.83">MATRIX13:83</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>
<b>end;</b></div>

<b>assume </b><a NAME="E5:69"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K3" title="MATRLIN2:func.3">AutEqMt</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/>

<a NAME="E6:69"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="matrlin2.html#E2:69"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="matrlin2.html#E3:69"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="matrix13.html#T83" title="MATRIX13:th.83">MATRIX13:83</a></i>;<br/>
<b>hence </b><a NAME="E7:69"/>
not  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c6">f</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="ref" href="matrlin2.html#T42" target="_self" title="MATRLIN2:th.42">Th42</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>
