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<b>let </b><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">X</font> be  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a> <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">V</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">VV</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c4">VX</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c5">AV</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c6">AX</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c7">MV</font> = the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c8">MX</font> = the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c9">SV</font> = the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c10">SX</font> = the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E2:21"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <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 <font color="Maroon" title="c2">X</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> <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 <font color="Maroon" title="c1">V</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</a></i>;<br/>
<a NAME="E3:21"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</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="c2">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green" title="E20">A3</font></i>: 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</a></i>;<br/>
<a NAME="E5:21"/>
( the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</a></i>;<br/>
<a NAME="E6:21"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E3:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></i>;<br/>
<a NAME="E7:21"/>
( the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c2">X</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E3:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></i>;<br/>
<a NAME="E9:21"/>
( the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">X</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="rusub_1.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:21"/>
<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="rusub_1.html#E3:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="rusub_1.html#E4:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="rusub_1.html#E6:21"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="rusub_1.html#E8:21"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></i>;<br/>


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