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<b>let </b><font color="Maroon">L</font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="polyalg1.html#L1" target="_self">AlgebraStr</a> of <font color="Maroon">L</font>, <font color="Maroon">B</font> be    <a href="polyalg1.html#L1" target="_self">AlgebraStr</a> of <font color="Maroon">L</font>, <font color="Maroon">C</font> be    <a href="polyalg1.html#L1" target="_self">AlgebraStr</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> is    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B</font> is    <a href="polyalg1.html#M1" target="_self">Subalgebra</a> of <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:33"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:33"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:33"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="polyalg1.html#D3" target="_self">POLYALG1:def 3</a><br/>

<b>thus </b> <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">A</font> = 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">B</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">C</font>
<b>by </b><i>, </i>
;<br/>

<b>thus </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">A</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">B</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">C</font>
<b>by </b><i>, </i>
;<br/>

<a NAME="E8:33"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">A</font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">B</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">C</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">C</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
;<br/>

<b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">A</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">B</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">C</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">C</font> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
;<br/>

<a NAME="E11:33"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>thus </b>the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">A</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">B</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">C</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">C</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>
;<br/>


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