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

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</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">V1</font> being   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">K</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="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font> holds  <font color="Olive" title="b2">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Olive" title="b2">f</font></span>)</span> is   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Olive" title="b2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Olive" title="b2">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font> be   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">K</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">  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="c2">V1</font> holds  <font color="Olive" title="b1">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Olive" title="b1">f</font></span>)</span> is   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c2">V1</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">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span> is   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">U</font> =  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E1:40"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ; <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="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:40_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:40_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:40_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font>
 
;<br/>
<a NAME="E7:40_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T9" onmouseover="rs('vectsp_4/T9')" onmouseout="rh()">VECTSP_4:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c6">x</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:40_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <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="c2">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="vectsp11.html#T24" target="_self" onmouseover="rs('vectsp11/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E11:40_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="vectsp11.html#T23" target="_self" onmouseover="rs('vectsp11/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" onmouseover="rs('vectsp11/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" onmouseover="rs('vectsp11/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/>
<a NAME="E13:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T24" target="_self" onmouseover="rs('vectsp11/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E14:40_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:40_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a></span>; <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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:40"/>
<font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span> is   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E9"><span class="lab"><font color="Green" title="E9">Lm1</font></span></a>, <a class="ref" href="funct_2.html#T6" onmouseover="rs('funct_2/T6')" onmouseout="rh()">FUNCT_2:6</a></span>; <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/>


</div>
