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<span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">S</font> being  <a href="ortsp_1.html#NM1" title="ORTSP_1:NM.1">OrtSp</a> of <font color="Maroon" title="c4">F</font><br/> for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st not <font color="Olive" title="b3">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/> <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">y</font>,<font color="Olive" title="b4">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">S</font> be   <a href="ortsp_1.html#NM1" title="ORTSP_1:NM.1">OrtSp</a> of <font color="Maroon" title="c4">F</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">b</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  st not <font color="Olive" title="b2">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/> <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>; <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="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  implies  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font></span><br/>

<div><span class="lab"><font color="Green" title="E35">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:54_1"/>
not <font color="Maroon" title="c9">y</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font></span><br/><a NAME="E2:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A3</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ortsp_1.html#T11" target="_self" title="ORTSP_1:th.11">Th11</a>, <a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a></span>;<br/><a NAME="E3:54_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="ortsp_1.html#T11" target="_self" title="ORTSP_1:th.11">Th11</a>, <a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a></span>;<br/><a NAME="E4:54_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c9">y</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="ortsp_1.html#D2" target="_self" title="ORTSP_1:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><span class="kw"> such that </span><br/><a NAME="E6:54_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
not <font color="Maroon" title="c7">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E7:54_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
not <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">and </span><br/><a NAME="E8:54_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
not <font color="Maroon" title="c9">y</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ;<br/><a NAME="E9:54_1"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
not <font color="Maroon" title="c10">r</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a></span>;<br/><a NAME="E10:54_1"/>
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="ortsp_1.html#D7" target="_self" title="ORTSP_1:def.7">Def7</a></span>;<br/><a NAME="E11:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/><a NAME="E12:54_1"/>
( not <font color="Maroon" title="c10">r</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp; not <font color="Maroon" title="c10">r</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a></span>;<br/><a NAME="E13:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E11:54_1"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="ortsp_1.html#T40" target="_self" title="ORTSP_1:th.40">Th40</a></span>;<br/><a NAME="E14:54_1"/>
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="ortsp_1.html#K1" title="ORTSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c9">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="ortsp_1.html#D7" target="_self" title="ORTSP_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E15:54_1"/>
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:54_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:54_2"/>
<font color="Maroon" title="c9">y</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font></span><br/><a NAME="E2:54_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c4">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="ortsp_1.html#T12" target="_self" title="ORTSP_1:th.12">Th12</a>, <a class="ref" href="ortsp_1.html#T44" target="_self" title="ORTSP_1:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_2"/>
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="ortsp_1.html#T44" target="_self" title="ORTSP_1:th.44">Th44</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">hence </span><a NAME="E4:54"/>
 <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ortsp_1.html#K2" title="ORTSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E35">A2</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/>


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