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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <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">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b4">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Olive" title="b3">y1</font>,<font color="Olive" title="b4">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">y2</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="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font> implies <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font> )</span><br/>



<b>assume </b><a NAME="E1:92"/>
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> or  not <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font> or <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<a NAME="E2:92"/><b>then </b><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T76" target="_self" title="CARD_3:th.76">Th76</a></i>;<br/>
<b>assume </b><a NAME="E3:92"/>
( <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</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">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font></span><br/>

<a NAME="E4:92"/><b>then </b><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c3">x2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T76" target="_self" title="CARD_3:th.76">Th76</a></i>;<br/>
<a NAME="E5:92"/>
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c5">y2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D4" title="FUNCT_4:def.4">FUNCT_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:92"/>
<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="card_3.html#E2:92"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="card_3.html#E4:92"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="card_3.html#T86" target="_self" title="CARD_3:th.86">Th86</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>
