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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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">V</font> being  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</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> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b1">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font> holds <br/><font color="Olive" title="b1">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Olive" title="b3">W2</font>,<font color="Olive" title="b2">W1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</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">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font> holds <br/><font color="Maroon" title="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Olive" title="b2">W2</font>,<font color="Olive" title="b1">W1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>; <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="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ; <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="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span><br/>

<a NAME="E2:54"/><span class="kw">then </span>
<font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_2.html#K1" title="RMOD_2:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D4" target="_self" title="RMOD_3:def.4">Def4</a></span>;<br/>
<a NAME="E3:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rmod_2.html#K1" title="RMOD_2:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#T18" target="_self" title="RMOD_3:th.18">Th18</a></span>;<br/>
<a NAME="E4:54"/>
 <a href="vectsp_2.html#G1" title="VECTSP_2:aggr.1">RightModStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U7</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="rmod_3.html#D4" target="_self" title="RMOD_3:def.4">Def4</a></span>;<br/>
<a NAME="E5:54"/><span class="kw">then </span>
 <a href="vectsp_2.html#G1" title="VECTSP_2:aggr.1">RightModStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U7</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c3">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rmod_3.html#E10"><span class="lab"><font color="Green" title="E5">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:54"/>
<font color="Maroon" title="c2">V</font> <a href="rmod_3.html#R1" title="RMOD_3:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="rmod_3.html#D4" target="_self" title="RMOD_3:def.4">Def4</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/>


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