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<b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">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="c3">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 <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b3">f2</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> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</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="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f2</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> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f2</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> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f2</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> )</span><br/><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">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="c3">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 <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b2">f2</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="c5">i</font>,<font color="Olive" title="b3">x</font></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">f1</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="c5">i</font>,<font color="Olive" title="b3">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f2</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="c5">i</font>,<font color="Olive" title="b3">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</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="c5">i</font>,<font color="Olive" title="b3">x</font></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">f1</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="c5">i</font>,<font color="Olive" title="b3">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f2</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="c5">i</font>,<font color="Olive" title="b3">x</font></span>)</span></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">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="c3">n</font></span>)</span>; <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">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 <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Olive" title="b1">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Olive" title="b1">x</font></span>)</span></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c8">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" 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="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> )</span><br/>











<a NAME="E1:47"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/>
<a NAME="E2:47"/><i><font color="Green" title="E35">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green" title="E36">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c5">i</font>,<font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:47"/><i><font color="Green" title="E37">A4</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:47_1"/>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:47"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="pdiff_1.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E2:47_1"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_1"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff ( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E4:47_1"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:47_1"/>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</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>
<a NAME="E5:47"/><i><font color="Green" title="E38">A5</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:47_2"/>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E2:47"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="pdiff_1.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E2:47_2"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_2"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff ( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E4:47_2"/><b>then </b>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:47_2"/>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 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>
<a NAME="E6:47"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/>
<a NAME="E7:47"/><b>then </b><i><font color="Green" title="E39">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E8:47"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E5:47"><i><font color="Green" title="E38">A5</font></i></a></i>;<br/>
<a NAME="E9:47"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E10:47"/><i><font color="Green" title="E41">A8</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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> <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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> )</span><br/>

<b>assume </b><a NAME="E1:47_3"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font></span><br/>

<a NAME="E2:47_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:47"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E3:47_3"/><b>then </b><i><font color="Green" title="E43">A10</font></i>: 
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_3"/><b>then </b><i><font color="Green" title="E44">A11</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<i><font color="Green" title="E45">A12</font></i>: <font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span> = 
<font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47_3"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_3"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_3"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<i><font color="Green" title="E46">A13</font></i>: <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span> = 
<font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47_3"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_3"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_3"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<a NAME="E7:47_3"/><i><font color="Green" title="E47">A14</font></i>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:47_3"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:47_3"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E7:47_3"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E7:47_3"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E7:47"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pdiff_1.html#E1:47_3"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="pdiff_1.html#E5:47_3"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="pdiff_1.html#E6:47_3"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:47_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E7:47"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pdiff_1.html#E1:47_3"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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>
<a NAME="E11:47"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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> <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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> )</span><br/>

<b>assume </b><a NAME="E1:47_4"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font></span><br/>

<a NAME="E2:47_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E2:47"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E3:47_4"/><b>then </b><i><font color="Green" title="E43">A16</font></i>: 
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_4"/><b>then </b><i><font color="Green" title="E44">A17</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<i><font color="Green" title="E45">A18</font></i>: <font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span> = 
<font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47_4"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_4"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_4"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<i><font color="Green" title="E46">A19</font></i>: <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span> = 
<font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E3:47_4"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_4"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E4:47_4"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<a NAME="E7:47_4"/><i><font color="Green" title="E47">A20</font></i>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:47_4"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:47_4"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E7:47_4"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E7:47_4"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E9:47"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pdiff_1.html#E1:47_4"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="pdiff_1.html#E5:47_4"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="pdiff_1.html#E6:47_4"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E9:47"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pdiff_1.html#E1:47_4"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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="E12:47"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</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="c5">i</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="pdiff_1.html#E7:47"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pdiff_1.html#E9:47"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pdiff_1.html#E10:47"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</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>
