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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">V</font> being  <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Olive" title="b1">V</font> holds  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Olive" title="b3">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Olive" title="b3">W2</font></span>)</span> <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="b3">W2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <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"> for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font> holds  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Olive" title="b2">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Olive" title="b2">W2</font></span>)</span> <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">W2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> <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="c4">W2</font></span><br/>







<span class="kw">thus </span><a NAME="E1:33"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> <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="c4">W2</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 ; <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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font></span><br/>

<a NAME="E2:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> : ( <font color="Olive" title="b1">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Olive" title="b2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D1" target="_self" title="RMOD_3:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v</font>
 <span class="kw">and </span><br/><a NAME="E5:33_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font>
 <span class="kw">and </span><br/><a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 ;<br/>
<a NAME="E7:33_1"/>
<font color="Maroon" title="c6">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="rmod_3.html#T7" target="_self" title="RMOD_3:th.7">Th7</a></span>;<br/>
<a NAME="E8:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="rmod_2.html#T28" title="RMOD_2:th.28">RMOD_2:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:33_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> )</span><br/>

<a NAME="E2:33"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rmod_3.html#E11"><span class="lab"><font color="Green" title="E6">Lm2</font></span></a></span>;<br/>
<a NAME="E3:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rmod_3.html#E10"><span class="lab"><font color="Green" title="E5">Lm1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:33"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E5:33"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E24">A4</font></span></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/>


</div>
