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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><br/> for <font color="Olive" title="b4">v</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</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/><span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Olive" title="b3">W2</font>,<font color="Olive" title="b2">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </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><br/> for <font color="Olive" title="b3">v</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</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/><span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Olive" title="b2">W2</font>,<font color="Olive" title="b1">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </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"> for <font color="Olive" title="b1">v</font> being  <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</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="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</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 <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  )</span><br/>









<span class="kw">assume </span><a NAME="E1:66"/><span class="lab"><font color="Green" title="E39">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span><br/>

<a NAME="E2:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D5" target="_self" title="RMOD_3:def.5">Def5</a></span>;<br/>
<a NAME="E3:66"/><span class="lab"><font color="Green" title="E41">A3</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="c4">W2</font>,<font color="Maroon" title="c3">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="rmod_3.html#T46" target="_self" title="RMOD_3:th.46">Th46</a></span>;<br/>
<a NAME="E4:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </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"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D5" target="_self" title="RMOD_3:def.5">Def5</a></span>;<br/>
<a NAME="E5:66"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="rmod_3.html#D5" target="_self" title="RMOD_3:def.5">Def5</a></span>;<br/>
<a NAME="E6:66"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </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"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="rmod_3.html#D5" target="_self" title="RMOD_3:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:66"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rmod_3.html#K4" title="RMOD_3:func.4">|--</a> <font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c3">W1</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="rmod_3.html#T51" target="_self" title="RMOD_3:th.51">Th51</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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