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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font>, <font color="Maroon" title="c5">W3</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>;<br/>









<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">W1</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c4">W2</font>
 ;<br/>

<b>thus </b><a NAME="E2:35"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><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></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:35_1"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_3.html#D2" target="_self" title="RMOD_3:def.2">Def2</a></i>;<br/>
<a NAME="E3:35_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:35_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : ( <font color="Olive" title="b1">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> &amp; <font color="Olive" title="b2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rmod_3.html#D1" target="_self" title="RMOD_3:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">u1</font>, <font color="Maroon" title="c8">v1</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E6:35_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v1</font>
 <b>and </b><br/><a NAME="E7:35_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c7">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 <b>and </b><br/><a NAME="E8:35_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W3</font>
 ;<br/>
<a NAME="E9:35_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c7">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E1:35"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="rmod_3.html#E7:35_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rmod_2.html#T16" title="RMOD_2:th.16">RMOD_2:16</a></i>;<br/>
<a NAME="E10:35_1"/><b>then </b><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c7">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <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>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E7:35_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rmod_3.html#T7" target="_self" title="RMOD_3:th.7">Th7</a></i>;<br/>
<a NAME="E11:35_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E2:35_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:35_1"/><b>then </b>
<font color="Maroon" title="c7">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E6:35_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E13:35_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c7">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E9:35_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="rmod_2.html#T31" title="RMOD_2:th.31">RMOD_2:31</a></i>;<br/>
<a NAME="E14:35_1"/><b>then </b>
<font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c7">u1</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<a NAME="E15:35_1"/><b>then </b>
<font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T66" title="VECTSP_1:th.66">VECTSP_1:66</a></i>;<br/>
<a NAME="E16:35_1"/><b>then </b>
<font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/>
<a NAME="E17:35_1"/><b>then </b>
<font color="Maroon" title="c8">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c5">W3</font>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E8:35_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="rmod_3.html#T7" target="_self" title="RMOD_3:th.7">Th7</a></i>;<br/>
<a NAME="E18:35_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rmod_3.html#E6:35_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="rmod_3.html#E10:35_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="rmod_3.html#T5" target="_self" title="RMOD_3:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E19:35_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:35"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rmod_3.html#E43"><i><font color="Green" title="E27">Lm10</font></i></a></i>;<br/>


</div>
