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

<a NAME="E1:160_1_2"/>
for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>  st ( for <font color="Olive" title="b3">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Olive" title="b1">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b3">t</font> ) &amp; ( for <font color="Olive" title="b3">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Olive" title="b2">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b3">t</font> ) holds <br/><font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="160_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>; <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">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">t</font> ) &amp; ( for <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">t</font> ) 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/>

<b>assume </b><b>that </b><br/><a NAME="E1:160_1_2_1"/><i><font color="Green" title="E131">B1</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">t</font>
 <b>and </b><br/><a NAME="E2:160_1_2_1"/><i><font color="Green" title="E132">B2</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E3:160_1_2_1"/>
 <a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="160_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:160_1_2_1_1"/>
for <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="160_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">t</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="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  implies <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> )</span><br/>

<b>assume </b><a NAME="E1:160_1_2_1_1_1"/><i><font color="Green" title="E133">C1</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span><br/>

<a NAME="E2:160_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">t</font>
 
<b>by </b><i><a class="txt" href="#E1:160_1_2_1"><i><font color="Green" title="E131">B1</font></i></a>, <a class="txt" href="#E1:160_1_2_1_1_1"><i><font color="Green" title="E133">C1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:160_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="#E2:160_1_2_1"><i><font color="Green" title="E132">B2</font></i></a>, <a class="txt" href="#E1:160_1_2_1_1_1"><i><font color="Green" title="E133">C1</font></i></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>
<b>hence </b><a NAME="E2:160_1_2_1_1"/>
 <a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c2">f1</font>,<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c3">f2</font>,<span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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>
<b>hence </b><a NAME="E4:160_1_2_1"/>
<font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="txt" href="modelc_2.html#E166"><i><font color="Green" title="E130">LemFid</font></i></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>
<b>hence </b><a NAME="E2:160_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <span class="p1">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span>  st ( for <font color="Olive" title="b3">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b3">t</font> ) &amp; ( for <font color="Olive" title="b3">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a>  holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="modelc_2.html#K27" title="MODELC_2:func.27">Inf_seq</a> <a href="modelc_2.html#K46" title="MODELC_2:func.46">AtomicFamily</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_2.html#K47" title="MODELC_2:func.47">AtomicFunc</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b3">t</font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font>
 ; <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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