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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">V</font> being  <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">V</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b2">V</font> holds <br/><font color="Olive" title="b2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">V</font> be   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">V</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c3">V</font> implies <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c3">V</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c2">X</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c5">VV</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>;<br/>
<a NAME="E3:34"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c3">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="rmod_2.html#D2" target="_self" title="RMOD_2:def.2">Def2</a></span>;<br/>
<a NAME="E4:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A3</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">AX</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">AV</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V</font>;<br/>
<a NAME="E5:34"/>
( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="c3">V</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="rmod_2.html#D2" target="_self" title="RMOD_2:def.2">Def2</a></span>;<br/>
<a NAME="E6:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A4</font></span>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">MX</font> = the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">MV</font> = the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c3">V</font>;<br/>
<a NAME="E7:34"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c3">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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="rmod_2.html#D2" target="_self" title="RMOD_2:def.2">Def2</a></span>;<br/>
<a NAME="E8:34"/>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">X</font> &amp; the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</a> of <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_2.html#U1" title="VECTSP_2:sel.1">rmult</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="rmod_2.html#D2" target="_self" title="RMOD_2:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:34"/>
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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