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

<span class="kw">set </span><font color="Maroon" title="c3">T</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="c4">TR</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">B</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="lab"><font color="Green" title="E42">A1</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> = 
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</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/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c2">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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="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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v2</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">P1</font> = <font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c10">P2</font> = <font color="Maroon" title="c8">v2</font>, <font color="Maroon" title="c11">FP1</font> = <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c12">FP2</font> = <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/><a NAME="E2:56_1_1_2"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
(  <a href="matrtop1.html#K1" title="MATRTOP1:func.1">@</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K2" title="MATRTOP1:func.2">@</a> <font color="Maroon" title="c11">FP1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">FP1</font> &amp;  <a href="matrtop1.html#K1" title="MATRTOP1:func.1">@</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K2" title="MATRTOP1:func.2">@</a> <font color="Maroon" title="c12">FP2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">FP2</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">p1</font> = <font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c14">p2</font> = <font color="Maroon" title="c8">v2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a></span>;<br/><a NAME="E4:56_1_1_2"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
(  <a href="matrtop1.html#K1" title="MATRTOP1:func.1">@</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K2" title="MATRTOP1:func.2">@</a> <font color="Maroon" title="c9">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P1</font> &amp;  <a href="matrtop1.html#K1" title="MATRTOP1:func.1">@</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K2" title="MATRTOP1:func.2">@</a> <font color="Maroon" title="c10">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P2</font> )
 ;<br/><font color="Maroon" title="c7">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">v2</font> = 
<font color="Maroon" title="c9">P1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c10">P2</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/>.= 
<font color="Maroon" title="c13">p1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_1_2"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="matrtop1.html#T1" title="MATRTOP1:th.1">MATRTOP1:1</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">p1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c11">FP1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c12">FP2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1_2"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="matrtop1.html#T1" title="MATRTOP1:th.1">MATRTOP1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v2</font></span>)</span>
<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 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="E5:56_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>;<br/>
<a NAME="E6:56_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">B</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" title="MATRTOP1:th.19">MATRTOP1:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">A</font> =  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> (<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c5">B</font>) as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c7">A</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">r</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <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">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <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> holds  <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c9">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">P</font> = <font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c12">FP</font> = <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font> = 
<font color="Maroon" title="c8">r</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c11">P</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/>.= 
<font color="Maroon" title="c8">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T17" title="MATRIXR1:th.17">MATRIXR1:17</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c9">v</font></span>)</span> = 
<font color="Maroon" title="c8">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#D4" title="TOPREAL9:def.4">TOPREAL9:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c12">FP</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T17" title="MATRIXR1:th.17">MATRIXR1:17</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font></span>)</span>
<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 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="E9:56_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D2" title="MOD_2:def.2">MOD_2:def 2</a></span>;<br/>
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c7">A</font> = 
 <a href="matrlin2.html#K4" title="MATRLIN2:func.4">Mx2Tran</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="c5">B</font>,<font color="Maroon" title="c5">B</font>)</span>)</span>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c5">B</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T20" title="MATRTOP1:th.20">MATRTOP1:20</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_1_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E5:56_1_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="matrlin2.html#T34" title="MATRLIN2:th.34">MATRLIN2:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:56_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" 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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