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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.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#NM1" title="PBOOLE:NM.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#M2" title="PBOOLE:mode.2">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#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>  st <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>  holds <br/><font color="Olive" title="b5">g</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="PBOOLE:NM.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#M2" title="PBOOLE:mode.2">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#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font>  st <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>  holds <br/><font color="Olive" title="b4">g</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/><span class="kw">let </span><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#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">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#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>  st <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>  holds <br/><font color="Olive" title="b2">g</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>  st <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>  holds <br/><font color="Olive" title="b1">g</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  implies <font color="Maroon" title="c6">g</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  )</span><br/>









<span class="kw">assume </span><a NAME="E1:2"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D3" onmouseover="rs('msualg_3/D3')" onmouseout="rh()">MSUALG_3:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I</font> or  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><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></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/>

<span class="kw">assume </span><a NAME="E2:2"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><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></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/>

<a NAME="E3:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A3</font></span>: 
( <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> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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></span>)</span> &amp; <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> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> = 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
<br/>.= 
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:2"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><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></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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="equation.html#T1" target="_self" onmouseover="rs('equation/T1')" onmouseout="rh()">Th1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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