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<span class="kw">let </span><font color="Maroon" title="c10">IT1</font>, <font color="Maroon" title="c11">IT2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  ex <font color="Olive" title="b1">p</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="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) &amp; (  for <font color="Olive" title="b1">q</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="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/><font color="Maroon" title="c10">IT1</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> ) &amp; (  for <font color="Olive" title="b1">q</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="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/><font color="Maroon" title="c11">IT2</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> ) implies <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font> ) &amp; ( (  for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) ) &amp; <font color="Maroon" title="c10">IT1</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> &amp; <font color="Maroon" title="c11">IT2</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> implies <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:45_1_2"/>
(  ex <font color="Olive" title="b1">p</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="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) &amp; (  for <font color="Olive" title="b1">q</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="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/><font color="Maroon" title="c10">IT1</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> ) &amp; (  for <font color="Olive" title="b1">q</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="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/><font color="Maroon" title="c11">IT2</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> ) implies <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) ) &amp; <font color="Maroon" title="c10">IT1</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> &amp; <font color="Maroon" title="c11">IT2</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> implies <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c12">p</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><a NAME="E2:45_1_2_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
(  not <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">q</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="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c10">IT1</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> ) or  ex <font color="Olive" title="b1">q</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="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c11">IT2</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span> ) or <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">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="E4:45_1_2_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
(  not <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</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="#E2:45_1_2_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:45_1_2_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
 for <font color="Olive" title="b1">q</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="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/><font color="Maroon" title="c10">IT1</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:45_1_2_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 for <font color="Olive" title="b1">q</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="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  &amp;  not <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  holds <br/><font color="Maroon" title="c11">IT2</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">q</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">y</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font></span><br/>

<a NAME="E7:45_1_2_1"/>
<font color="Maroon" title="c10">IT1</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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c13">r</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c13">r</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c6">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="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> (<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c13">r</font>,<font color="Maroon" title="c7">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_2_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E4:45_1_2_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:45_1_2_1"/>
<font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_1_2_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E4:45_1_2_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E2:45_1_2"/>
( (  for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>  holds <br/>( <font color="Maroon" title="c8">a</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  or <font color="Maroon" title="c6">x</font> <a href="symsp_1.html#NR1" title="SYMSP_1:NR.1">_|_</a>  ) ) &amp; <font color="Maroon" title="c10">IT1</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> &amp; <font color="Maroon" title="c11">IT2</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> implies <font color="Maroon" title="c10">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">IT2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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