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<b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) &amp; ( for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) implies <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3_1_2"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>and </b><br/><a NAME="E2:3_1_2"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><a NAME="E2:3_1_2_1"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="ref" href="msscyc_2.html#T1" target="_self" title="MSSCYC_2:th.1">Th1</a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><b>hence </b><a NAME="E3:3_1_2_1"/>
 <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:3_1_2"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/><b>assume </b><a NAME="E1:3_1_2_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msscyc_2.html#K1" title="MSSCYC_2:func.1">InducedEdges</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><a NAME="E2:3_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; <font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="txt" href="#E1:3_1_2"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="#E2:3_1_2"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:3_1_2_2"/>
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:3_1_2"/>
<font color="Maroon" title="c4">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">F2</font>
 <b>by </b><i><a class="txt" href="#E4:3_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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