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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> 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="b2">W1</font>, <font color="Olive" title="b3">W2</font>, <font color="Olive" title="b4">W3</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">W1</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b4">W3</font> &amp; <font color="Olive" title="b3">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b4">W3</font> holds <br/><font color="Olive" title="b2">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b3">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b4">W3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</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">W1</font>, <font color="Olive" title="b2">W2</font>, <font color="Olive" title="b3">W3</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">W1</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b3">W3</font> &amp; <font color="Olive" title="b2">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b3">W3</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Olive" title="b2">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b3">W3</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font>, <font color="Maroon" title="c5">W3</font> be   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</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="c3">W1</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</font> &amp; <font color="Maroon" title="c4">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</font> implies <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
( <font color="Maroon" title="c3">W1</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</font> &amp; <font color="Maroon" title="c4">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</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">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font> holds <br/><font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</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="c6">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c6">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font> )</span><br/><span class="kw">assume </span><a NAME="E1:51_1"/>
<font color="Maroon" title="c6">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</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="c6">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:51_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
( <font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 <span class="kw">and </span><br/><a NAME="E4:51_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T92" onmouseover="rs('zmodul01/T92')" onmouseout="rh()">ZMODUL01:92</a></span>;<br/><a NAME="E5:51_1"/>
( <font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="zmodul01.html#T23" onmouseover="rs('zmodul01/T23')" onmouseout="rh()">ZMODUL01:23</a></span>;<br/><span class="kw">hence </span><a NAME="E6:51_1"/>
<font color="Maroon" title="c6">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="zmodul01.html#T36" onmouseover="rs('zmodul01/T36')" onmouseout="rh()">ZMODUL01:36</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="E3:51"/>
<font color="Maroon" title="c3">W1</font> <a href="zmodul01.html#K1" title="ZMODUL01:func.1">+</a> <font color="Maroon" title="c4">W2</font> is   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c5">W3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T44" onmouseover="rs('zmodul01/T44')" onmouseout="rh()">ZMODUL01:44</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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