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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">Seqts1</font>, <font color="Maroon" title="c3">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="c2">Seqts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Seqts2</font> implies <span class="p1">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</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="c2">Seqts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</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="c3">Seqts2</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c2">Seqts1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c3">Seqts2</font>;<br/><span class="kw">assume </span><a NAME="E1:85_1_1_1"/>
<font color="Maroon" title="c2">Seqts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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#K24" title="FOMODEL4:func.24">R2</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="c2">Seqts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</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="c3">Seqts2</font></span><br/><span class="kw">set </span><font color="Maroon" title="c6">R</font> =  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c1">S</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">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="85_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#K24" title="FOMODEL4:func.24">R2</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="c2">Seqts1</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#K24" title="FOMODEL4:func.24">R2</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="c3">Seqts2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:85_1_1_1_1"/><span class="lab"><font color="Green" title="E69">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#K24" title="FOMODEL4:func.24">R2</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="c2">Seqts1</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#K24" title="FOMODEL4:func.24">R2</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="c3">Seqts2</font></span><br/><a NAME="E2:85_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">C0</font></span>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">Seqts1</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel4.html#K10" title="FOMODEL4:func.10">P2</a> <font color="Maroon" title="c1">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E73"><span class="lab"><font color="Green" title="E68">Lm1e</font></span></a></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="#E1:85_1_1_1_1"><span class="lab"><font color="Green" title="E69">CC0</font></span></a></span>;<br/><a NAME="E4:85_1_1_1_1"/>
<font color="Maroon" title="c9">seqt</font> <a href="fomodel4.html#R3" title="FOMODEL4:pred.3">Rule2</a> <font color="Maroon" title="c2">Seqts1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D33" target="_self" title="FOMODEL4:def.33">DefP2</a>, <a class="txt" href="#E2:85_1_1_1_1"><span class="lab"><font color="Green" title="E70">C0</font></span></a></span>;<br/><a NAME="E5:85_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">seqt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  ex <font color="Olive" title="b1">t</font> being   <a href="fomodel1.html#V13" title="FOMODEL1:attr.13">termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> st <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#K34" title="FOMODEL2:func.34">TheEqSymbOf</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="Olive" title="b1">t</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Olive" title="b1">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D20" target="_self" title="FOMODEL4:def.20">Def2</a></span>;<br/><a NAME="E6:85_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">seqt</font> <a href="fomodel4.html#R3" title="FOMODEL4:pred.3">Rule2</a> <font color="Maroon" title="c3">Seqts2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D20" target="_self" title="FOMODEL4:def.20">Def2</a></span>;<br/><a NAME="E7:85_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="c3">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#K10" title="FOMODEL4:func.10">P2</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D33" target="_self" title="FOMODEL4:def.33">DefP2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:85_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#K24" title="FOMODEL4:func.24">R2</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="c3">Seqts2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E70"><span class="lab"><font color="Green" title="E65">Lm1</font></span></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="E3:85_1_1_1"/>
<span class="p1">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</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="c2">Seqts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</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="c3">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="E2:85_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#K24" title="FOMODEL4:func.24">R2</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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