<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</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">  for <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b3">T</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b4">E1</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b3">T</font>)<br/>  for <font color="Olive" title="b5">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">E1</font>  st <font color="Olive" title="b4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/> <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b3">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b5">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><br/>  for <font color="Olive" title="b2">T</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><br/>  for <font color="Olive" title="b3">E1</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b2">T</font>)<br/>  for <font color="Olive" title="b4">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">E1</font>  st <font color="Olive" title="b3">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b2">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b4">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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">T</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><br/>  for <font color="Olive" title="b2">E1</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>)<br/>  for <font color="Olive" title="b3">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">E1</font>  st <font color="Olive" title="b2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/> <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b3">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">T</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">E</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">E1</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)<br/>  for <font color="Olive" title="b2">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">E1</font>  st <font color="Olive" title="b1">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/> <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b2">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">E1</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</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">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">E1</font>  st <font color="Maroon" title="c5">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/> <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Olive" title="b1">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">b1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">E1</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"> ( <font color="Maroon" title="c5">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) )</span><br/>











<span class="kw">assume </span><a NAME="E1:79"/><span class="lab"><font color="Green" title="E45">AS</font></span>: 
( <font color="Maroon" title="c5">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</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">  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/>

<a NAME="E2:79"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/>
<a NAME="E3:79"/>
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="79_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font> holds <br/><font color="Olive" title="b1">o</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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font> implies <font color="Olive">b<sub>1</sub></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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:79_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</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"> <font color="Olive">b<sub>1</sub></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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> or <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_1_1_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</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"> <font color="Olive">b<sub>1</sub></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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span><br/><div class="add"><a NAME="E2:79_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">B1</font></span>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:79_1_1_1_1"/><span class="lab"><font color="Green" title="E48">B2</font></span>: 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T35" onmouseover="rs('field_6/T35')" onmouseout="rh()">FIELD_6:35</a></span>;<br/><a NAME="E4:79_1_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E45">AS</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:79_1_1_1_1"/>
<font color="Maroon" title="c7">o</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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1_1"><span class="lab"><font color="Green" title="E47">B1</font></span></a>, <a class="txt" href="#E3:79_1_1_1_1"><span class="lab"><font color="Green" title="E48">B2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_1_1_1_2"/><span class="lab"><font color="Green" title="E47">B1</font></span>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</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"> <font color="Olive">b<sub>1</sub></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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span><br/><div class="add"><a NAME="E2:79_1_1_1_2"/><span class="lab"><font color="Green" title="E48">B2</font></span>: 
<font color="Maroon" title="c4">T</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T35" onmouseover="rs('field_6/T35')" onmouseout="rh()">FIELD_6:35</a></span>;<br/><a NAME="E3:79_1_1_1_2"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>) is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T36" onmouseover="rs('field_6/T36')" onmouseout="rh()">FIELD_6:36</a></span>;<br/><a NAME="E4:79_1_1_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</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 <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:79_1_1_1_2"/>
<font color="Maroon" title="c7">o</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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1_2"><span class="lab"><font color="Green" title="E48">B2</font></span></a>, <a class="txt" href="#E1:79_1_1_1_2"><span class="lab"><font color="Green" title="E47">B1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:79_1"/>
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</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"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A</font></span>: 
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>) is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E45">AS</font></span></a>, <a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="field_6.html#T37" onmouseover="rs('field_6/T37')" onmouseout="rh()">FIELD_6:37</a></span>;<br/>
<a NAME="E5:79"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>) is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T10" target="_self" onmouseover="rs('field_7/T10')" onmouseout="rh()">ext0</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:79"/>
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2">(<span class="default"><span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T35" onmouseover="rs('field_6/T35')" onmouseout="rh()">FIELD_6:35</a></span>;<br/>
<a NAME="E7:79"/><span class="kw">then </span>
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></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 <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2">(<span class="default"><span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E45">AS</font></span></a>, <a class="ref" href="xboole_1.html#T11" onmouseover="rs('xboole_1/T11')" onmouseout="rh()">XBOOLE_1:11</a></span>;<br/>
<a NAME="E8:79"/><span class="kw">then </span>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E45">AS</font></span></a>, <a class="ref" href="field_6.html#T37" onmouseover="rs('field_6/T37')" onmouseout="rh()">FIELD_6:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:79"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1">(<span class="default"><span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c4">T</font>)</span>)</span>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">b1</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E47">A</font></span></a>, <a class="ref" href="ec_pf_1.html#T4" onmouseover="rs('ec_pf_1/T4')" onmouseout="rh()">EC_PF_1:4</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/>


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