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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V1</font>, <font color="Olive" title="b2">V2</font>, <font color="Olive" title="b3">V3</font> being  <a href="matrlin.html#V1" title="MATRLIN:attr.1">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="b4">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><br/> for <font color="Olive" title="b5">g</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V2</font>,<font color="Olive" title="b3">V3</font>  st <font color="Olive" title="b5">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Olive" title="b4">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/>(  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Olive" title="b4">f</font> &amp;  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Olive" title="b4">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</font>, <font color="Maroon" title="c4">V3</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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><br/> for <font color="Olive" title="b2">g</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V2</font>,<font color="Maroon" title="c4">V3</font>  st <font color="Olive" title="b2">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Olive" title="b1">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/>(  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Olive" title="b1">f</font> &amp;  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V2</font>,<font color="Maroon" title="c4">V3</font>  st <font color="Olive" title="b1">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/>(  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c5">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V2</font>,<font color="Maroon" title="c4">V3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies (  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c5">f</font> ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:71"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c5">f</font> )</span><br/>

the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#D2" title="RANKNULL:def.2">RANKNULL:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrlin2.html#E61"><span class="lab"><font color="Green" title="E39">Lm8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#D2" title="RANKNULL:def.2">RANKNULL:def 2</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#D2" title="RANKNULL:def.2">RANKNULL:def 2</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E41">A2</font></span>:  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></span>
<br/>.= 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c5">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="ranknull.html#T45" title="RANKNULL:th.45">RANKNULL:45</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c5">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#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T44" title="RANKNULL:th.44">RANKNULL:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T44" title="RANKNULL:th.44">RANKNULL:44</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:71"/>
(  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
