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

<span class="kw">let </span><font color="Maroon" title="c1">Fml</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Fml</font><br/>  for <font color="Olive" title="b2">R</font> being    <a href="proofs_1.html#M1" title="PROOFS_1:mode.1">Rule</a> of <font color="Maroon" title="c1">Fml</font><br/>  for <font color="Olive" title="b3">P</font> being  <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a>  <a href="proofs_1.html#NM4" title="PROOFS_1:NM.4">Formula-sequence</a> holds  <font color="Olive" title="b3">P</font> is    <a href="proofs_1.html#M4" title="PROOFS_1:mode.4">Formula-sequence</a> of <font color="Maroon" title="c1">Fml</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Fml</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being    <a href="proofs_1.html#M1" title="PROOFS_1:mode.1">Rule</a> of <font color="Maroon" title="c1">Fml</font><br/>  for <font color="Olive" title="b2">P</font> being  <font color="Maroon" title="c2">B</font>,<font color="Olive">b<sub>1</sub></font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a>  <a href="proofs_1.html#NM4" title="PROOFS_1:NM.4">Formula-sequence</a> holds  <font color="Olive" title="b2">P</font> is    <a href="proofs_1.html#M4" title="PROOFS_1:mode.4">Formula-sequence</a> of <font color="Maroon" title="c1">Fml</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be    <a href="proofs_1.html#M1" title="PROOFS_1:mode.1">Rule</a> of <font color="Maroon" title="c1">Fml</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being  <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a>  <a href="proofs_1.html#NM4" title="PROOFS_1:NM.4">Formula-sequence</a> holds  <font color="Olive" title="b1">P</font> is    <a href="proofs_1.html#M4" title="PROOFS_1:mode.4">Formula-sequence</a> of <font color="Maroon" title="c1">Fml</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font> be  <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a>  <a href="proofs_1.html#NM4" title="PROOFS_1:NM.4">Formula-sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">P</font> is    <a href="proofs_1.html#M4" title="PROOFS_1:mode.4">Formula-sequence</a> of <font color="Maroon" title="c1">Fml</font></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c3">R</font></span>)</span>;<br/>
<a NAME="E1:56"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Fml</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#D2" target="_self" onmouseover="rs('proofs_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E2:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Fml</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E3:56"/>
 <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Fml</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <font color="Maroon" title="c4">P</font> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Fml</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <font color="Maroon" title="c4">P</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Fml</font></span><br/>

<a NAME="E2:56_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c6">t</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:56_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Fml</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="proofs_1.html#E37"><span class="lab"><font color="Green" title="E25">Lm54</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>
<a NAME="E4:56"/><span class="kw">then </span>
<font color="Maroon" title="c4">P</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Fml</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56"/>
<font color="Maroon" title="c4">P</font> is    <a href="proofs_1.html#M4" title="PROOFS_1:mode.4">Formula-sequence</a> of <font color="Maroon" title="c1">Fml</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#D5" target="_self" onmouseover="rs('proofs_1/D5')" onmouseout="rh()">Def5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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