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<span class="kw">set </span><font color="Maroon" title="c2">R</font> =  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">Q</font> = <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">Seqts</font>, <font color="Maroon" title="c5">Seqts2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Seqts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Seqts2</font> implies <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">Seqts2</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c4">Seqts</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c5">Seqts2</font>;<br/><span class="kw">assume </span><a NAME="E1:139_1_1_1"/><span class="lab"><font color="Green" title="E94">B0</font></span>: 
<font color="Maroon" title="c4">Seqts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Seqts2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">Seqts2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">Seqts2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:139_1_1_1_1"/><span class="lab"><font color="Green" title="E95">CC0</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">Seqts2</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">seqt</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139_1_1_1_1"><span class="lab"><font color="Green" title="E95">CC0</font></span></a></span>;<br/><a NAME="E3:139_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">Seqts</font>,<font color="Maroon" title="c9">seqt</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel4.html#K19" title="FOMODEL4:func.19">P8</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139_1_1_1_1"><span class="lab"><font color="Green" title="E95">CC0</font></span></a>, <a class="txt" href="fomodel4.html#E73"><span class="lab"><font color="Green" title="E68">Lm1e</font></span></a></span>;<br/><a NAME="E4:139_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">seqt</font> <a href="fomodel4.html#R12" title="FOMODEL4:pred.12">Rule8</a> <font color="Maroon" title="c4">Seqts</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D42" target="_self" title="FOMODEL4:def.42">DefP8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c12">phi</font>, <font color="Maroon" title="c13">phi1</font>, <font color="Maroon" title="c14">phi2</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E6:139_1_1_1_1"/><span class="lab"><font color="Green" title="E96">C1</font></span>: 
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Seqts</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Seqts</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c10">y1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">phi1</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c13">phi1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c14">phi2</font> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seqt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c9">seqt</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c12">phi</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c12">phi</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D29" target="_self" title="FOMODEL4:def.29">Def8</a></span>;<br/><a NAME="E7:139_1_1_1_1"/>
<font color="Maroon" title="c9">seqt</font> <a href="fomodel4.html#R12" title="FOMODEL4:pred.12">Rule8</a> <font color="Maroon" title="c5">Seqts2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D29" target="_self" title="FOMODEL4:def.29">Def8</a>, <a class="txt" href="#E6:139_1_1_1_1"><span class="lab"><font color="Green" title="E96">C1</font></span></a>, <a class="txt" href="#E1:139_1_1_1"><span class="lab"><font color="Green" title="E94">B0</font></span></a></span>;<br/><a NAME="E8:139_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">Seqts2</font>,<font color="Maroon" title="c9">seqt</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel4.html#K19" title="FOMODEL4:func.19">P8</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D42" target="_self" title="FOMODEL4:def.42">DefP8</a></span>;<br/><span class="kw">hence </span><a NAME="E9:139_1_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">Seqts2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#T2" target="_self" title="FOMODEL4:th.2">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:139_1_1_1"/>
<span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">Seqts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">Seqts2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:139_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="fomodel4.html#NM1" title="FOMODEL4:NM.1">Rule</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="fomodel4.html#V5" title="FOMODEL4:attr.5">isotone</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D9" target="_self" title="FOMODEL4:def.9">DefMonotonic1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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