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<span class="kw">let </span><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span> holds  <font color="Olive" title="b2">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b4">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b4">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span> holds  <font color="Olive" title="b1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b3">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b3">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span> holds  <font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b2">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b2">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</font></span>)</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">  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</span> holds  <font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c4">m</font></span>)</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"> <font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span><br/>











<a NAME="E1:47"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D4" onmouseover="rs('vfunct_1/D4')" onmouseout="rh()">VFUNCT_1:def 4</a></span>;<br/>
<a NAME="E2:47"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D4" onmouseover="rs('vfunct_1/D4')" onmouseout="rh()">VFUNCT_1:def 4</a></span>;<br/>
<a NAME="E3:47"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:47"/>
 for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</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"> ( <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> )</span><br/>

<a NAME="E1:47_1"/>
( <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:47_1"/>
( <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</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/>


</div><span class="kw">end;</span></div>
<a NAME="E5:47"/><span class="kw">then </span>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/>
<a NAME="E6:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<a NAME="E7:47"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</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"> ( <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> )</span><br/>

<a NAME="E1:47_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:47_2"/>
( <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> iff <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</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/>


</div><span class="kw">end;</span></div>
<a NAME="E8:47"/>
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>  st <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</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"> ( <font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_3"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font></span><br/>

<a NAME="E2:47_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D4" onmouseover="rs('vfunct_1/D4')" onmouseout="rh()">VFUNCT_1:def 4</a></span>;<br/>
<a NAME="E3:47_3"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: <font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font></span>)</span> = 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_3"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_3"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/>
<span class="lab"><font color="Green" title="E36">A10</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E4:47_3"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="vfunct_1.html#D4" onmouseover="rs('vfunct_1/D4')" onmouseout="rh()">VFUNCT_1:def 4</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:47_3"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:47_3"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="vfunct_1.html#D4" onmouseover="rs('vfunct_1/D4')" onmouseout="rh()">VFUNCT_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:47_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_3"><span class="lab"><font color="Green" title="E36">A10</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:47"/>
<font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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/>


</div>
