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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   non 2 <a href="ring_3.html#V3" title="RING_3:attr.3">-characteristic</a>   <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>   non  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</a>  <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> holds <br/> ( <font color="Olive" title="b1">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_9.html#V5" title="FIELD_9:attr.5">-quadratic</a>  iff  ex <font color="Olive" title="b2">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Olive" title="b1">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font> )</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"> ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_9.html#V5" title="FIELD_9:attr.5">-quadratic</a>  iff  ex <font color="Olive" title="b1">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font> )</span><br/>



<a NAME="E1:155"/><span class="lab"><font color="Green" title="E69">H1</font></span>: 
( <font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a>, <a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/>
<a NAME="E2:155"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">H2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</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 <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E71">A</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_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"> ( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_9.html#V5" title="FIELD_9:attr.5">-quadratic</a>  implies  ex <font color="Olive" title="b1">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:155_1"/><span class="lab"><font color="Green" title="E72">AS</font></span>: 
<font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_9.html#V5" title="FIELD_9:attr.5">-quadratic</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">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">K</font> = <font color="Maroon" title="c2">E</font> as  <font color="Maroon" title="c1">F</font> <a href="field_4.html#V2" title="FIELD_4:attr.2">-finite</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_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">  not <!--HUMANRECFAILED--> for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> holds  <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:155_1_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> holds  <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> contradiction</span><br/><a NAME="E2:155_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</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 <font color="Maroon" title="c1">F</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:155_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1"><span class="lab"><font color="Green" title="E72">AS</font></span></a>, <a class="txt" href="#E2:155"><span class="lab"><font color="Green" title="E70">H2</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a>, <a class="ref" href="field_7.html#T7" onmouseover="rs('field_7/T7')" onmouseout="rh()">FIELD_7:7</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">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E5:155_1"/><span class="lab"><font color="Green" title="E73">A</font></span>: 
 not <font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
 ;<br/><a NAME="E6:155_1"/><span class="lab"><font color="Green" title="E74">B</font></span>: 
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:155_1_2"/>
<font color="Maroon" title="c3">K</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a> 
 
<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="E2:155_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">C1</font></span>: 
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1"><span class="lab"><font color="Green" title="E72">AS</font></span></a>, <a class="ref" href="field_5.html#T15" onmouseover="rs('field_5/T15')" onmouseout="rh()">FIELD_5:15</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_2_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">  not  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:155_1_2_1"/>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"> contradiction</span><br/><a NAME="E2:155_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">C2</font></span>: 
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:155_1_2_1"/>
<span class="p1">(<span class="default"><a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E4:155_1_2_1"/><span class="kw">then </span>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:155_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">C3</font></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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_2_1"><span class="lab"><font color="Green" title="E76">C2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="ref" href="field_7.html#T7" onmouseover="rs('field_7/T7')" onmouseout="rh()">FIELD_7:7</a></span>;<br/><a NAME="E6:155_1_2_1"/><span class="lab"><font color="Green" title="E78">C4</font></span>: 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="E7:155_1_2_1"/>
<font color="Maroon" title="c4">a</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="c4">a</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:155_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:155_1"><span class="lab"><font color="Green" title="E73">A</font></span></a>, <a class="txt" href="#E5:155_1_2_1"><span class="lab"><font color="Green" title="E77">C3</font></span></a>, <a class="txt" href="#E6:155_1_2_1"><span class="lab"><font color="Green" title="E78">C4</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">hence </span><a NAME="E4:155_1_2"/>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_2"><span class="lab"><font color="Green" title="E75">C1</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:155_1"/><span class="kw">then </span>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T67" onmouseover="rs('field_6/T67')" onmouseout="rh()">FIELD_6:67</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> =  <a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">F</font>) as   <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>   <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#D4" target="_self" onmouseover="rs('field_9/D4')" onmouseout="rh()">defquadr</a></span>;<br/><a NAME="E9:155_1"/><span class="lab"><font color="Green" title="E75">D</font></span>: 
<font color="Maroon" title="c3">K</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:155_1_3"/><span class="lab"><font color="Green" title="E76">D1</font></span>: 
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>) is  <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D8" onmouseover="rs('field_4/D8')" onmouseout="rh()">FIELD_4:def 8</a></span>;<br/>
<span class="lab"><font color="Green" title="E77">D2</font></span>:  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> = 
2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1"><span class="lab"><font color="Green" title="E72">AS</font></span></a>, <a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>
<br/>.= 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1"><span class="lab"><font color="Green" title="E74">B</font></span></a>, <a class="ref" href="field_4.html#D7" onmouseover="rs('field_4/D7')" onmouseout="rh()">FIELD_4:def 7</a></span>
;<br/>
<a NAME="E3:155_1_3"/><span class="lab"><font color="Green" title="E78">D3</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a> 
 
<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="E4:155_1_3"/>
 <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:155_1_3"><span class="lab"><font color="Green" title="E78">D3</font></span></a>, <a class="ref" href="field_5.html#T14" onmouseover="rs('field_5/T14')" onmouseout="rh()">FIELD_5:14</a></span>;<br/>
<a NAME="E5:155_1_3"/><span class="kw">then </span>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1_3"><span class="lab"><font color="Green" title="E76">D1</font></span></a>, <a class="txt" href="#E2:155_1_3"><span class="lab"><font color="Green" title="E77">D2</font></span></a>, <a class="ref" href="vectsp_9.html#T28" onmouseover="rs('vectsp_9/T28')" onmouseout="rh()">VECTSP_9:28</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E79">D4</font></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_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p2">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D4" onmouseover="rs('vectsp_4/D4')" onmouseout="rh()">VECTSP_4:def 4</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D4" onmouseover="rs('vectsp_4/D4')" onmouseout="rh()">VECTSP_4:def 4</a></span>
<br/>.= 
 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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>
;<br/>
<a NAME="E7:155_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">D8</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/>
<a NAME="E8:155_1_3"/><span class="lab"><font color="Green" title="E81">D6</font></span>: 
<font color="Maroon" title="c3">K</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_3_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:155_1_3_3"/>
(  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">K</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> &amp;  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> <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="c3">K</font> )
 
<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="E2:155_1_3_3"/>
<font color="Maroon" title="c3">K</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="#E7:155_1_3"><span class="lab"><font color="Green" title="E80">D8</font></span></a>, <a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c3">K</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">K</font>,<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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="#E6:155_1_3"><span class="lab"><font color="Green" title="E79">D4</font></span></a>, <a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/>
<a NAME="E10:155_1_3"/><span class="lab"><font color="Green" title="E82">D7</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:155_1_3"><span class="lab"><font color="Green" title="E81">D6</font></span></a>, <a class="txt" href="field_9.html#E111"><span class="lab"><font color="Green" title="E68">unique20</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_3_4"><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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> holds  <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155"><span class="lab"><font color="Green" title="E70">H2</font></span></a></span>;<br/><a NAME="E2:155_1_3_4"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:155_1_3_4"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</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/></div><span class="kw">end;</span></div>
<a NAME="E12:155_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> is <font color="Maroon" title="c1">F</font> <a href="field_8.html#V1" title="FIELD_8:attr.1">-fixing</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E13:155_1_3"/>
<font color="Maroon" title="c3">K</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:155_1_3"><span class="lab"><font color="Green" title="E82">D7</font></span></a>, <a class="ref" href="field_8.html#D5" onmouseover="rs('field_8/D5')" onmouseout="rh()">FIELD_8:def 5</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">consider </span><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E11:155_1"/><span class="lab"><font color="Green" title="E76">E</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T22" target="_self" onmouseover="rs('field_9/T22')" onmouseout="rh()">qua5a</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c9">c1</font> = <font color="Maroon" title="c7">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155"><span class="lab"><font color="Green" title="E70">H2</font></span></a></span>;<br/><a NAME="E13:155_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</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="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T10" onmouseover="rs('field_4/T10')" onmouseout="rh()">FIELD_4:10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">p1</font> = <font color="Maroon" title="c5">p</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c3">K</font></span>)</span> ;<br/><a NAME="E15:155_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/><a NAME="E16:155_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">F</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:155_1"><span class="lab"><font color="Green" title="E76">E</font></span></a>, <a class="ref" href="field_9.html#T23" target="_self" onmouseover="rs('field_9/T23')" onmouseout="rh()">eval2</a></span>;<br/> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> = 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T51" onmouseover="rs('field_6/T51')" onmouseout="rh()">FIELD_6:51</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>
;<br/><a NAME="E18:155_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c10">p1</font>
 ;<br/><a NAME="E19:155_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">G</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c10">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D10" onmouseover="rs('polynom5/D10')" onmouseout="rh()">POLYNOM5:def 10</a></span>;<br/><span class="lab"><font color="Green" title="E79">G1</font></span>: <font color="Maroon" title="c8">b1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  = 
<font color="Maroon" title="c8">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
;<br/><a NAME="E21:155_1"/>
4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c9">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T7" target="_self" onmouseover="rs('field_9/T7')" onmouseout="rh()">Z3</a></span>;<br/><a NAME="E22:155_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c9">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/><a NAME="E23:155_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">U2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c9">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:155_1"><span class="lab"><font color="Green" title="E79">G1</font></span></a>, <a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><span class="lab"><font color="Green" title="E81">I</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#T5" onmouseover="rs('realalg2/T5')" onmouseout="rh()">REALALG2:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c9">c1</font></span>)</span>

;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E26:155_1"/><span class="lab"><font color="Green" title="E82">U1</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c9">c1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:155_1"><span class="lab"><font color="Green" title="E78">G</font></span></a>, <a class="txt" href="#E16:155_1"><span class="lab"><font color="Green" title="E77">F</font></span></a>, <a class="ref" href="field_9.html#T26" target="_self" onmouseover="rs('field_9/T26')" onmouseout="rh()">lemeval2</a>, <a class="ref" href="o_ring_1.html#D2" onmouseover="rs('o_ring_1/D2')" onmouseout="rh()">O_RING_1:def 2</a></span>;<br/><a NAME="E27:155_1"/><span class="lab"><font color="Green" title="E83">L0</font></span>: 
 not <font color="Maroon" title="c3">K</font> is 2 <a href="ring_3.html#V3" title="RING_3:attr.3">-characteristic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a></span>;<br/><a NAME="E28:155_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">L1</font></span>: 
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c9">c1</font>,<font color="Maroon" title="c8">b1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p5">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p5">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:155_1"><span class="lab"><font color="Green" title="E81">I</font></span></a>, <a class="txt" href="#E26:155_1"><span class="lab"><font color="Green" title="E82">U1</font></span></a>, <a class="ref" href="field_9.html#T27" target="_self" onmouseover="rs('field_9/T27')" onmouseout="rh()">TC0</a></span>;<br/><a NAME="E29:155_1"/><span class="lab"><font color="Green" title="E85">G5</font></span>: 
 not 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:155_1"><span class="lab"><font color="Green" title="E83">L0</font></span></a>, <a class="ref" href="field_9.html#T5" target="_self" onmouseover="rs('field_9/T5')" onmouseout="rh()">ch2</a></span>;<br/> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font> = 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T51" onmouseover="rs('field_6/T51')" onmouseout="rh()">FIELD_6:51</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>
;<br/><a NAME="E31:155_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c10">p1</font>
 ;<br/><a NAME="E32:155_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">L2</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c10">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D10" onmouseover="rs('polynom5/D10')" onmouseout="rh()">POLYNOM5:def 10</a></span>;<br/><a NAME="E33:155_1"/><span class="lab"><font color="Green" title="E87">L3</font></span>: 
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_8"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_8_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="E1:155_1_8_1"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:155_1"><span class="lab"><font color="Green" title="E86">L2</font></span></a>, <a class="txt" href="#E16:155_1"><span class="lab"><font color="Green" title="E77">F</font></span></a>, <a class="txt" href="#E28:155_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:155_1_8_1_1"/>
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:155_1"><span class="lab"><font color="Green" title="E85">G5</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:155_1_8_1_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:155_1_8_1_2"/>
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:155_1"><span class="lab"><font color="Green" title="E85">G5</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:155_1_8_1_2"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E34:155_1"/><span class="lab"><font color="Green" title="E88">U3</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_9"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:155_1_9"/><span class="lab"><font color="Green" title="E89">G2</font></span>: 
( <font color="Maroon" title="c3">K</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  &amp; <font color="Maroon" title="c3">K</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  )
 
<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="E2:155_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">G3</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> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) &amp; <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> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) &amp;  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c3">K</font> &amp;  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c3">K</font> )
 
<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>, <a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/>
<a NAME="E3:155_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">I1</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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/>
<a NAME="E4:155_1_9"/><span class="lab"><font color="Green" title="E92">I2</font></span>: 
( <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> &amp; <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="E5:155_1_9"/>
<font color="Maroon" title="c11">u</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="c11">u</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">b2</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c13">c2</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c14">u2</font> = <font color="Maroon" title="c11">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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="#E3:155_1_9"><span class="lab"><font color="Green" title="E91">I1</font></span></a>, <a class="txt" href="#E4:155_1_9"><span class="lab"><font color="Green" title="E92">I2</font></span></a></span>;<br/>
<a NAME="E7:155_1_9"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_9_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:155_1_9_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<a NAME="E2:155_1_9_1"/><span class="kw">then </span>
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1_9"><span class="lab"><font color="Green" title="E89">G2</font></span></a>, <a class="ref" href="field_9.html#T7" target="_self" onmouseover="rs('field_9/T7')" onmouseout="rh()">Z3</a></span>;<br/>
<a NAME="E3:155_1_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">G4</font></span>: 
<span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:155_1"><span class="lab"><font color="Green" title="E85">G5</font></span></a>, <a class="ref" href="field_6.html#T18" onmouseover="rs('field_6/T18')" onmouseout="rh()">FIELD_6:18</a></span>;<br/>
<a NAME="E4:155_1_9_1"/><span class="lab"><font color="Green" title="E94">G6</font></span>: 
(  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font> &amp;  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">u2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_9_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="E1:155_1_9_1_1"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:155_1"><span class="lab"><font color="Green" title="E86">L2</font></span></a>, <a class="txt" href="#E16:155_1"><span class="lab"><font color="Green" title="E77">F</font></span></a>, <a class="txt" href="#E28:155_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:155_1_9_1_1_1"/><span class="lab"><font color="Green" title="E95">K</font></span>: 
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )</span><br/><div class="add"><a NAME="E2:155_1_9_1_1_1"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:155_1_9_1"><span class="lab"><font color="Green" title="E94">G6</font></span></a>, <a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:155_1_9_1_1_1"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1_9_1_1_1"><span class="lab"><font color="Green" title="E95">K</font></span></a>, <a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="txt" href="#E3:155_1_9_1"><span class="lab"><font color="Green" title="E93">G4</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</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="155_1_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:155_1_9_1_1_2"/><span class="lab"><font color="Green" title="E95">K</font></span>: 
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )</span><br/><div class="add"><a NAME="E2:155_1_9_1_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:155_1_9_1"><span class="lab"><font color="Green" title="E94">G6</font></span></a>, <a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:155_1_9_1_1_2"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">b2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1_9_1_1_2"><span class="lab"><font color="Green" title="E95">K</font></span></a>, <a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="txt" href="#E3:155_1_9_1"><span class="lab"><font color="Green" title="E93">G4</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</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>
<a NAME="E8:155_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R2" title="TARSKI: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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 
;<br/>
<a NAME="E9:155_1_9"/><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="c4">a</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:155_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">G4</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</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="E11:155_1_9"/>
<font color="Maroon" title="c4">a</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="c4">a</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">b2</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c16">c2</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c17">a2</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="#E3:155_1_9"><span class="lab"><font color="Green" title="E91">I1</font></span></a>, <a class="txt" href="#E4:155_1_9"><span class="lab"><font color="Green" title="E92">I2</font></span></a></span>;<br/>
<a NAME="E13:155_1_9"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p5">(<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="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_9_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:155_1_9_2"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<a NAME="E2:155_1_9_2"/><span class="kw">then </span>
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155_1_9"><span class="lab"><font color="Green" title="E89">G2</font></span></a>, <a class="ref" href="field_9.html#T7" target="_self" onmouseover="rs('field_9/T7')" onmouseout="rh()">Z3</a></span>;<br/>
<a NAME="E3:155_1_9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">G6</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<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="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>;<br/>
<a NAME="E4:155_1_9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">G5</font></span>: 
<font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_9_2_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="E1:155_1_9_2_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:155_1"><span class="lab"><font color="Green" title="E87">L3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_9_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:155_1_9_2_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</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"> ( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p5">(<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="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:155_1_9_2_1_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p5">(<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="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:155_1_9_2"><span class="lab"><font color="Green" title="E94">G6</font></span></a>, <a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</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="155_1_9_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:155_1_9_2_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span></span>)</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"> ( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p5">(<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="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:155_1_9_2_1_2"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p4">(<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="p5"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">b2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p5">(<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="p0"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</font></span></a>, <a class="txt" href="#E4:155_1_9_2"><span class="lab"><font color="Green" title="E95">G5</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</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>
<a NAME="E14:155_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="tarski.html#R2" title="TARSKI: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> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 
;<br/>
<a NAME="E15:155_1_9"/><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="c11">u</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:155_1_9"/><span class="kw">then </span>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</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="#E2:155_1_9"><span class="lab"><font color="Green" title="E90">G3</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="E17:155_1_9"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">u</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="#E10:155_1_9"><span class="lab"><font color="Green" title="E93">G4</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/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E89">U4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_1_10"><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">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  holds  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:155_1_10"/>
 ex <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E3:155_1_10"/><span class="lab"><font color="Green" title="E90">G7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a> 
 ;<br/><a NAME="E4:155_1_10"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/><a NAME="E5:155_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">F2</font></span>: 
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T7" target="_self" onmouseover="rs('field_9/T7')" onmouseout="rh()">Z3</a></span>;<br/><a NAME="E6:155_1_10"/>
 not 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:155_1"><span class="lab"><font color="Green" title="E83">L0</font></span></a>, <a class="ref" href="field_9.html#T5" target="_self" onmouseover="rs('field_9/T5')" onmouseout="rh()">ch2</a></span>;<br/><a NAME="E7:155_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">F3</font></span>: 
<span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E5:155_1_10"><span class="lab"><font color="Green" title="E91">F2</font></span></a>, <a class="ref" href="field_6.html#T18" onmouseover="rs('field_6/T18')" onmouseout="rh()">FIELD_6:18</a></span>;<br/><a NAME="E8:155_1_10"/><span class="lab"><font color="Green" title="E93">F4</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_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="E1:155_1_10_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w</font> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">w</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:155_1_10"><span class="lab"><font color="Green" title="E90">G7</font></span></a>, <a class="txt" href="#E26:155_1"><span class="lab"><font color="Green" title="E82">U1</font></span></a>, <a class="txt" href="#E23:155_1"><span class="lab"><font color="Green" title="E80">U2</font></span></a>, <a class="ref" href="field_9.html#T8" target="_self" onmouseover="rs('field_9/T8')" onmouseout="rh()">lemquadr</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:155_1_10_1_1"/><span class="lab"><font color="Green" title="E94">S</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w</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"> contradiction</span><br/><div class="add"><a NAME="E2:155_1_10_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E8:155_1_10"><span class="lab"><font color="Green" title="E93">F4</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><a NAME="E3:155_1_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">F5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E7:155_1_10"><span class="lab"><font color="Green" title="E92">F3</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>;<br/><a NAME="E4:155_1_10_1_1"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E1:155_1_10_1_1"><span class="lab"><font color="Green" title="E94">S</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/><a NAME="E5:155_1_10_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E8:155_1_10"><span class="lab"><font color="Green" title="E93">F4</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><a NAME="E6:155_1_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">F6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E7:155_1_10"><span class="lab"><font color="Green" title="E92">F3</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_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="E1:155_1_10_1_1_1"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:155_1"><span class="lab"><font color="Green" title="E86">L2</font></span></a>, <a class="txt" href="#E16:155_1"><span class="lab"><font color="Green" title="E77">F</font></span></a>, <a class="txt" href="#E28:155_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:155_1_10_1_1_1_1"/>
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:155_1_10_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:155_1_10_1_1"><span class="lab"><font color="Green" title="E95">F5</font></span></a>, <a class="txt" href="#E5:155_1"><span class="lab"><font color="Green" title="E73">A</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="155_1_10_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:155_1_10_1_1_1_2"/>
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:155_1_10_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_10_1_1"><span class="lab"><font color="Green" title="E96">F6</font></span></a>, <a class="txt" href="#E5:155_1"><span class="lab"><font color="Green" title="E73">A</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:155_1_10_1_2"/><span class="lab"><font color="Green" title="E94">S</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">w</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"> contradiction</span><br/><div class="add"><a NAME="E2:155_1_10_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E8:155_1_10"><span class="lab"><font color="Green" title="E93">F4</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><a NAME="E3:155_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">F5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">w</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E7:155_1_10"><span class="lab"><font color="Green" title="E92">F3</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>;<br/><a NAME="E4:155_1_10_1_2"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E1:155_1_10_1_2"><span class="lab"><font color="Green" title="E94">S</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/><a NAME="E5:155_1_10_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E8:155_1_10"><span class="lab"><font color="Green" title="E93">F4</font></span></a>, <a class="ref" href="field_6.html#T15" onmouseover="rs('field_6/T15')" onmouseout="rh()">FIELD_6:15</a></span>;<br/><a NAME="E6:155_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">F6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:155"><span class="lab"><font color="Green" title="E69">H1</font></span></a>, <a class="txt" href="#E7:155_1_10"><span class="lab"><font color="Green" title="E92">F3</font></span></a>, <a class="ref" href="field_6.html#T16" onmouseover="rs('field_6/T16')" onmouseout="rh()">FIELD_6:16</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_1_2_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="E1:155_1_10_1_2_1"/>
( <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> or <font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:155_1"><span class="lab"><font color="Green" title="E86">L2</font></span></a>, <a class="txt" href="#E16:155_1"><span class="lab"><font color="Green" title="E77">F</font></span></a>, <a class="txt" href="#E28:155_1"><span class="lab"><font color="Green" title="E84">L1</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="155_1_10_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:155_1_10_1_2_1_1"/>
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:155_1_10_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:155_1_10_1_2"><span class="lab"><font color="Green" title="E95">F5</font></span></a>, <a class="txt" href="#E5:155_1"><span class="lab"><font color="Green" title="E73">A</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="155_1_10_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:155_1_10_1_2_1_2"/>
<font color="Maroon" title="c4">a</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">u</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:155_1_10_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_10_1_2"><span class="lab"><font color="Green" title="E96">F6</font></span></a>, <a class="txt" href="#E5:155_1"><span class="lab"><font color="Green" title="E73">A</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">v</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c7">c</font></span>)</span> as   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D2" onmouseover="rs('o_ring_1/D2')" onmouseout="rh()">O_RING_1:def 2</a></span>;<br/><a NAME="E37:155_1"/><span class="lab"><font color="Green" title="E90">F5</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c12">v</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c12">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T58" target="_self" onmouseover="rs('field_9/T58')" onmouseout="rh()">Fi3a</a></span>;<br/><a NAME="E38:155_1"/><span class="lab"><font color="Green" title="E91">F6</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"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c12">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:155_1"><span class="lab"><font color="Green" title="E82">U1</font></span></a>, <a class="txt" href="#E23:155_1"><span class="lab"><font color="Green" title="E80">U2</font></span></a>, <a class="txt" href="#E34:155_1"><span class="lab"><font color="Green" title="E88">U3</font></span></a>, <a class="txt" href="#E35:155_1"><span class="lab"><font color="Green" title="E89">U4</font></span></a>, <a class="ref" href="field_9.html#T71" target="_self" onmouseover="rs('field_9/T71')" onmouseout="rh()">m102</a></span>;<br/><a NAME="E39:155_1"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>), <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c12">v</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:155_1"><span class="lab"><font color="Green" title="E90">F5</font></span></a>, <a class="txt" href="#E38:155_1"><span class="lab"><font color="Green" title="E91">F6</font></span></a>, <a class="ref" href="field_8.html#T58" onmouseover="rs('field_8/T58')" onmouseout="rh()">FIELD_8:58</a></span>;<br/><span class="kw">hence </span><a NAME="E40:155_1"/>
 ex <font color="Olive" title="b1">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:155_1"><span class="lab"><font color="Green" title="E75">D</font></span></a>, <a class="ref" href="field_8.html#T44" onmouseover="rs('field_8/T44')" onmouseout="rh()">FIELD_8:44</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="155_2"><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"> (  ex <font color="Olive" title="b1">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font> implies  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c3">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><a NAME="E2:155_2"/><span class="lab"><font color="Green" title="E72">A</font></span>: 
<font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <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">  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/><a NAME="E3:155_2"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>),<font color="Maroon" title="c2">E</font> <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_2"><span class="lab"><font color="Green" title="E72">A</font></span></a>, <a class="ref" href="field_8.html#T43" onmouseover="rs('field_8/T43')" onmouseout="rh()">FIELD_8:43</a></span>;<br/><span class="kw">hence </span> <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>) = 
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_8.html#T45" onmouseover="rs('field_8/T45')" onmouseout="rh()">FIELD_8:45</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_9.html#E84"><span class="lab"><font color="Green" title="E53">dega</font></span></a></span>
;<br/> <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">hence </span><a NAME="E5:155"/>
( <font color="Maroon" title="c2">E</font> is <font color="Maroon" title="c1">F</font> <a href="field_9.html#V5" title="FIELD_9:attr.5">-quadratic</a>  iff  ex <font color="Olive" title="b1">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">E</font>, <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_8.html#R1" title="FIELD_8:pred.1">are_isomorphic_over</a> <font color="Maroon" title="c1">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:155"><span class="lab"><font color="Green" title="E71">A</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>
