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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">M</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b2">W2</font>;<br/>
<a NAME="E1:67_1_1"/><i><font color="Green" title="E47">A1</font></i>: 
for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">M</font>  ex <font color="Olive" title="b3">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">M</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font>,<font color="Olive" title="b3">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">M</font>;<br/>



<b>consider </b><font color="Maroon" title="c5">W1</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E2:67_1_1_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c5">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">W2</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E4:67_1_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c6">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c5">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c6">W2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">C</font>
;<br/>

<b>thus </b><a NAME="E6:67_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">A2</font>,<font color="Maroon" title="c7">C</font>]
 <b>by </b><i><a class="txt" href="vectsp_5.html#E2:67_1_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="vectsp_5.html#E4:67_1_1_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:67_1_1"/>
 ex <font color="Olive" title="b1">o</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">M</font> st <br/>for <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">M</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">A1</font>,<font color="Olive" title="b3">A2</font>,<font color="Olive" title="b1">o</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">A1</font>,<font color="Olive" title="b3">A2</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="vectsp_5.html#E1:67_1_1"><i><font color="Green" title="E47">A1</font></i></a>);</i><br/>


</div>
