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<span class="kw">let </span><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">L</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> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/>  for <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="c2">L</font><br/>  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b1">V</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</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">W</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Olive" title="b1">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Olive" title="b1">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Olive" title="b1">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">W</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c2">L</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">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>);<br/>
<span class="kw">thus </span><a NAME="E1:71"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">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"> <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</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="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span><br/>

<a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</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="c4">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<a NAME="E2:71_1"/>
( <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</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="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <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="c4">W</font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</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="c4">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:71_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</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="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K6" title="GRCAT_1:func.6">ZeroMap</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</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>


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