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

<span class="kw">let </span><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c2">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b3">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span>  ) &amp; ( for <font color="Olive" title="b1">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c3">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b3">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span>  ) implies <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_2"/><span class="lab"><font color="Green" title="E3">A17</font></span>: 
for <font color="Olive" title="b1">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c2">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b3">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> st not <font color="Maroon" title="c3">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b3">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span>  or <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:12_1_2"/><span class="lab"><font color="Green" title="E4">A18</font></span>: 
for <font color="Olive" title="b1">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c3">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b3">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> holds <br/><font color="Olive" title="b3">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font></span><br/>

<a NAME="E3:12_1_2"/><span class="lab"><font color="Green" title="E5">A19</font></span>: 
for <font color="Olive" title="b1">O</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> holds <br/><font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">O</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_2_1"/>
<font color="Maroon" title="c4">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">O</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">O</font> = <font color="Maroon" title="c4">O</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<font color="Maroon" title="c2">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">O</font> = 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font> holds <br/><font color="Olive" title="b2">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> </span>}</span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_2"><span class="lab"><font color="Green" title="E3">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">O</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_2"><span class="lab"><font color="Green" title="E4">A18</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:12_1_2_1"/>
<font color="Maroon" title="c2">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:12_1_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:12_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_2"><span class="lab"><font color="Green" title="E5">A19</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
