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<span class="kw">let </span><font color="Maroon" title="c1">i</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">V1</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V1</font>,<font color="Olive" title="b2">V1</font> holds   <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">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   <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V1</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c3">V1</font> holds   <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c3">V1</font>; <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#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c4">f</font></span><br/>







<a NAME="E1:35"/>
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#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</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="c4">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">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="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/>
<font color="Maroon" title="c5">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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 ; <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="c5">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="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">v</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> ;<br/>
<a NAME="E3:35_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c3">V1</font> &amp; <font color="Maroon" title="c6">v</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">V1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T14" title="RANKNULL:th.14">RANKNULL:14</a>, <a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></span>;<br/>
<a NAME="E4:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T24" target="_self" title="VECTSP11:th.24">Th24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_1"/>
<font color="Maroon" title="c5">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="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:35"/>
 <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="vectsp11.html#K2" title="VECTSP11:func.2">|^</a> <font color="Maroon" title="c1">i</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp11.html#K3" title="VECTSP11:func.3">UnionKers</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</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>
