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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">w</font>, <font color="Olive" title="b6">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b3">u</font>,<font color="Olive" title="b4">v</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b6">y</font> holds <br/><font color="Olive" title="b3">u</font>,<font color="Olive" title="b4">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b6">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font>, <font color="Maroon" title="c7">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">and </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">and </span><br/><a NAME="E2:11_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c6">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:11_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">u</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:11_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E6:11_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="analoaf.html#T7" target="_self" title="ANALOAF:th.7">Th16</a></span>;<br/><a NAME="E7:11_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="xreal_1.html#T122" title="XREAL_1:th.122">XREAL_1:122</a></span>;<br/><a NAME="E8:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A9</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E10:11_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:11_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">and </span><br/><a NAME="E12:11_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:11_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="analoaf.html#T7" target="_self" title="ANALOAF:th.7">Th16</a></span>;<br/><span class="lab"><font color="Green" title="E24">A13</font></span>: <font color="Maroon" title="c3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">d</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E11:11_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="analoaf.html#T6" target="_self" title="ANALOAF:th.6">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/><a NAME="E14:11_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:11_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="xreal_1.html#T122" title="XREAL_1:th.122">XREAL_1:122</a></span>;<br/><a NAME="E15:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:11_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>;<br/><font color="Maroon" title="c3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E5:11_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="analoaf.html#T6" target="_self" title="ANALOAF:th.6">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E17:11_1"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E15:11_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:11"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D1" target="_self" title="ANALOAF:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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