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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</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">V</font>, <font color="Olive" title="b2">W</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b3">T</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font><br/>  for <font color="Olive" title="b4">A</font>, <font color="Olive" title="b5">B</font>, <font color="Olive" title="b6">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b4">A</font> <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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b3">T</font></span>)</span> &amp; <font color="Olive" title="b6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">A</font> holds <br/> <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b6">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b5">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</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">T</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">A</font> <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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b1">T</font></span>)</span> &amp; <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">A</font> holds <br/> <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b3">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</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">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">A</font> <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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font></span>)</span> &amp; <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">A</font> holds <br/> <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Olive" title="b2">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</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="c5">A</font> <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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font></span>)</span> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A</font> implies  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c6">B</font></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c5">A</font> <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="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">A</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">  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c6">B</font></span>)</span></span><br/>

<a NAME="E3:46"/><span class="lab"><font color="Green" title="E25">P1</font></span>: 
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c5">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="relat_1.html#T120" onmouseover="rs('relat_1/T120')" onmouseout="rh()">RELAT_1:120</a></span>;<br/>
<span class="kw">thus </span> <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c7">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E25">P1</font></span></a>, <a class="ref" href="mod_3.html#T12" onmouseover="rs('mod_3/T12')" onmouseout="rh()">MOD_3:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c3">W</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T22" target="_self" onmouseover="rs('zmodul07/T22')" onmouseout="rh()">LMTh441</a>, <a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>
<br/>.= 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c6">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T9" onmouseover="rs('vectsp_5/T9')" onmouseout="rh()">VECTSP_5:9</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/>


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