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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> holds <br/> (  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Olive" title="b1">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  iff  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  iff  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">MM</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">nV</font> = <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">mV</font> = <font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Bn</font> =  <a href="matrix13.html#K11" title="MATRIX13:func.11">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,<font color="Maroon" title="c1">n</font>)</span>)</span> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T45" title="MATRLIN2:th.45">MATRLIN2:45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">Bm</font> =  <a href="matrix13.html#K11" title="MATRIX13:func.11">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,<font color="Maroon" title="c2">m</font>)</span>)</span> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T45" title="MATRLIN2:th.45">MATRLIN2:45</a></span>;<br/>
<span class="lab"><font color="Green" title="E31">A1</font></span>:  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> = 
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" title="EUCLID:th.22">EUCLID:22</a></span>
;<br/>
<a NAME="E4:57"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">Bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T19" target="_self" title="MATRTOP1:th.19">Th19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">M1</font> = <font color="Maroon" title="c3">M</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">Bn</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">Bm</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T19" target="_self" title="MATRTOP1:th.19">Th19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">T</font> =  <a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>);<br/>
<a NAME="E6:57"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A3</font></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"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a></span>
;<br/>
<a NAME="E8:57"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T20" target="_self" title="MATRTOP1:th.20">Th20</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> implies  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:57_3"/>
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/><a NAME="E2:57_3"/><span class="kw">then </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"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><span class="kw">then </span> <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span> = 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T112" title="MATRIX13:th.112">MATRIX13:112</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:57_3"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T49" title="MATRLIN2:th.49">MATRLIN2:49</a></span>; <a class="txt" 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>

<a NAME="E10:57"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T112" title="MATRIX13:th.112">MATRIX13:112</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:57"/>
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c2">m</font> = 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T49" title="MATRLIN2:th.49">MATRLIN2:49</a></span>
<br/>.= 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span></span>)</span>

;<br/>
<a NAME="E13:57"/><span class="kw">then </span>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</a> (<font color="Maroon" title="c9">M1</font>,<font color="Maroon" title="c7">Bn</font>,<font color="Maroon" title="c8">Bm</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="vectsp_9.html#T28" title="VECTSP_9:th.28">VECTSP_9:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:57"/>
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c3">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
