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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b4">g1</font>, <font color="Olive" title="b5">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">Y</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g1</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f1</font> holds <br/><font color="Olive" title="b4">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">Y</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f1</font> holds <br/><font color="Olive" title="b3">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">Y</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>  st <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f1</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">f1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">Y</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> implies <font color="Maroon" title="c6">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font></span><br/>

<span class="lab"><font color="Green" title="E16">A2</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g1</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>
;<br/>
<a NAME="E3:19"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">Y</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="relset_1.html#T5" title="RELSET_1:th.5">RELSET_1:5</a></span>;<br/>
<a NAME="E4:19"/>
 for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:19"/>
<font color="Maroon" title="c6">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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