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<div class="add">

<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> being  <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="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#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b2">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">V1</font> be   <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">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#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>





<b>set </b><font color="Maroon" title="c4">U</font> =  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E1:40"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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> )
 <b>by </b><i><a class="txt" href="#E1:40_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:40_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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:40_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:40_1"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:40_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c6">x</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V1</font> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:40_1"/><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</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>
 <b>by </b><i><a class="txt" href="#E5:40_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="vectsp11.html#T24" target="_self" title="VECTSP11:th.24">Th24</a></i>;<br/>
<a NAME="E10:40_1"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<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#K2" title="VECTSP11:func.2">|^</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>
<b>by </b><i><a class="txt" href="#E9:40_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="vectsp11.html#T23" target="_self" title="VECTSP11:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</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>
<b>by </b><i><a class="ref" href="vectsp11.html#T20" target="_self" title="VECTSP11:th.20">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="matrlin2.html#K10" title="MATRLIN2:func.10">*</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>
<b>by </b><i><a class="ref" href="vectsp11.html#T19" target="_self" title="VECTSP11:th.19">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</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>
<b>by </b><i><a class="txt" href="#E10:40_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E12:40_1"/><b>then </b>
( <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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E3:40_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="vectsp11.html#T24" target="_self" title="VECTSP11:th.24">Th24</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E13:40_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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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>hence </b><a NAME="E2:40"/>
<font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">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#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp11.html#E9"><i><font color="Green" title="E9">Lm1</font></i></a>, <a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</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>
