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<b>let </b><font color="Maroon">o1</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font>, <font color="Maroon">o2</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:62_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">A1</font>, <font color="Olive">A2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font><br/> for <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> <font color="Olive">W2</font> holds <br/><font color="Maroon">o1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A1</font>,<font color="Olive">A2</font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Olive">W2</font>
 ;<br/>

<b>assume </b><a NAME="E2:62_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">A1</font>, <font color="Olive">A2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font><br/> for <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font>  st <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> <font color="Olive">W2</font> holds <br/><font color="Maroon">o2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A1</font>,<font color="Olive">A2</font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Olive">W2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:62_1_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font>, <font color="Maroon">B</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font> ;<br/><b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:62_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</a>, </i>;<br/><b>consider </b><font color="Maroon">W2</font> being  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:62_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E7:62_1_2_1"/>
( <font color="Maroon">o1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">o2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font> )
 <b>by </b><i>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a>, <a class="txt" href="rmod_3.html#E10"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:62_1_2_1"/>
<font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1_2"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T1">BINOP_1:1</a></i>;<br/>


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