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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <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">w2</font>, <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">u2</font>, <font color="Olive" title="b5">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b6">a2</font>, <font color="Olive" title="b7">c2</font>, <font color="Olive" title="b8">A3</font>, <font color="Olive" title="b9">A3'</font>, <font color="Olive" title="b10">B3</font>, <font color="Olive" title="b11">B3'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b4">u2</font></span>)</span> &amp; <font color="Olive" title="b5">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b10">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b8">A3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b10">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">c2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b4">u2</font></span>)</span> &amp; <font color="Olive" title="b10">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b11">B3'</font> &amp; <font color="Olive" title="b8">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b11">B3'</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b6">a2</font> &amp; <font color="Olive" title="b10">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">A3'</font> holds <br/><font color="Olive" title="b5">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">w2</font></span><br/><b>let </b><font color="Maroon" title="c2">w2</font>, <font color="Maroon" title="c3">o</font>, <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">u2</font>, <font color="Maroon" title="c6">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</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">a2</font>, <font color="Olive" title="b2">c2</font>, <font color="Olive" title="b3">A3</font>, <font color="Olive" title="b4">A3'</font>, <font color="Olive" title="b5">B3</font>, <font color="Olive" title="b6">B3'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">A3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">c2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Olive" title="b5">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">B3'</font> &amp; <font color="Olive" title="b3">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">B3'</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">a2</font> &amp; <font color="Olive" title="b5">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">A3'</font> holds <br/><font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">w2</font></span><br/><b>let </b><font color="Maroon" title="c7">a2</font>, <font color="Maroon" title="c8">c2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <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">A3</font>, <font color="Olive" title="b2">A3'</font>, <font color="Olive" title="b3">B3</font>, <font color="Olive" title="b4">B3'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Olive" title="b3">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B3'</font> &amp; <font color="Olive" title="b1">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B3'</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a2</font> &amp; <font color="Olive" title="b3">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A3'</font> holds <br/><font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">w2</font></span><br/><b>let </b><font color="Maroon" title="c9">A3</font>, <font color="Maroon" title="c10">A3'</font>, <font color="Maroon" title="c11">B3</font>, <font color="Maroon" title="c12">B3'</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B3'</font> &amp; <font color="Maroon" title="c9">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B3'</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a2</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A3'</font> implies <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">w2</font> )</span><br/>

















<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">A3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span> &amp; <font color="Maroon" title="c11">B3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B3'</font> &amp; <font color="Maroon" title="c9">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B3'</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a2</font> &amp; <font color="Maroon" title="c11">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A3'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">w2</font></span><br/>

<b>hence </b><font color="Maroon" title="c6">w1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">c2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c11">B3</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">w2</font>
<b>by </b><i><a class="txt" href="anproj_2.html#E1:42"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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