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<b>let </b><font color="Maroon" title="c1">I</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"> for <font color="Olive" title="b1">A</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b4">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b5">G</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>  st <font color="Olive" title="b4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b5">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> holds <br/><font color="Olive" title="b5">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">B</font>, <font color="Olive" title="b2">C</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font><br/> for <font color="Olive" title="b4">G</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>  st <font color="Olive" title="b3">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b4">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> holds <br/><font color="Olive" title="b4">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a></span><br/><b>let </b><font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>  st <font color="Olive" title="b1">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b2">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> holds <br/><font color="Olive" title="b2">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a></span><br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</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">G</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>  st <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b1">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> holds <br/><font color="Olive" title="b1">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a></span><br/><b>let </b><font color="Maroon" title="c6">G</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c6">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> implies <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c6">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</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">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</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="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/><b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <b>by </b><i><a class="txt" href="autalg_1.html#E1:21_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:21_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:21_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><a NAME="E5:21_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:21_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><a NAME="E6:21_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="autalg_1.html#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="autalg_1.html#E1:21_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/><a NAME="E7:21_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E4:21_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="autalg_1.html#E5:21_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="funct_2.html#T20" title="FUNCT_2:th.20">FUNCT_2:20</a></i>;<br/><b>hence </b><a NAME="E8:21_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:21_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</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="E4:21"/>
<font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 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/>


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