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<b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">VS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">h</font> being   <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><br/> for <font color="Olive" title="b2">H</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> holds <br/><font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H</font> ) &amp; ( for <font color="Olive" title="b1">h</font> being   <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><br/> for <font color="Olive" title="b2">H</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> holds <br/><font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H</font> ) implies <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_2"/><i><font color="Green" title="E3">A6</font></i>: 
for <font color="Olive" title="b1">h1</font> being   <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><br/> for <font color="Olive" title="b2">H1</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font>  st <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H1</font>
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green" title="E4">A7</font></i>: 
for <font color="Olive" title="b1">h2</font> being   <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><br/> for <font color="Olive" title="b2">H2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font>  st <font color="Olive" title="b1">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H2</font> holds <br/><font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font></span><br/>

<a NAME="E3:6_1_2"/>
for <font color="Olive" title="b1">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font> holds <br/><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font> implies <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font> )</span><br/>

<b>assume </b><a NAME="E1:6_1_2_1"/><i><font color="Green" title="E5">A8</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font></span><br/>

<a NAME="E2:6_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">h</font> is    <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">H</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E4:6_1_2_1"/><i><font color="Green" title="E6">A9</font></i>: 
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font>
 ;<br/>
<a NAME="E5:6_1_2_1"/>
<font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:6_1_2"><i><font color="Green" title="E3">A6</font></i></a>, <a class="txt" href="vectsp_8.html#E1:6_1_2_1"><i><font color="Green" title="E5">A8</font></i></a>, <a class="txt" href="vectsp_8.html#E4:6_1_2_1"><i><font color="Green" title="E6">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_2_1"/>
<font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">h</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E2:6_1_2"><i><font color="Green" title="E4">A7</font></i></a>, <a class="txt" href="vectsp_8.html#E1:6_1_2_1"><i><font color="Green" title="E5">A8</font></i></a>, <a class="txt" href="vectsp_8.html#E4:6_1_2_1"><i><font color="Green" title="E6">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_2"/>
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
