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

<b>set </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font>;<br/>
<b>set </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font>;<br/>
<b>set </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font>;<br/>
<a NAME="E1:14_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</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> <font color="Maroon" title="c4">B</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:14_1_1"/>
<font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<b>assume </b><a NAME="E1:14_1_1_1"/>
<font color="Maroon" title="c12">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span>
 ; <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"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<a NAME="E2:14_1_1_1"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:14_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) &amp; (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:14_1_1_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c8">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ) )
 
<b>by </b><i><a class="txt" href="circcomb.html#E1:14_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="circcomb.html#E2:14_1_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:14_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is    <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"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">h</font> = <font color="Maroon" title="c7">h1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c8">h2</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">B</font> ;<br/>
<a NAME="E4:14_1_1"/>
<font color="Maroon" title="c12">h</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M6</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>) )</span><br/>

<b>assume </b><a NAME="E1:14_1_1_2"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M6</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>)</span><br/>

<a NAME="E2:14_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) &amp; (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:14_1_1_2"/><b>then </b>
( ( <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> ) or ( <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c8">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c6">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="c13">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> ) )
 
<b>by </b><i><a class="txt" href="circcomb.html#E1:14_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="circcomb.html#E1:14_1_1_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_1_2"/>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M6</a>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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="E5:14_1_1"/>
<font color="Maroon" title="c7">h1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">h2</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">f1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c3">g1</font> <a href="circcomb.html#K1" title="CIRCCOMB:func.1">+*</a> <font color="Maroon" title="c6">g2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
