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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">X1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f1</font> &amp; <font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f2</font> &amp; <font color="Olive" title="b3">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <font color="Olive" title="b4">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">X1</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b4">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b4">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">X1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  st <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Olive" title="b1">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X1</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &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#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font> &amp; <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X1</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:28"/>
( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  or  not <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X1</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  or ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  ) )</span><br/>

<a NAME="E2:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:28"/>
( <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X1</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  )</span><br/>

<a NAME="E4:28"/><span class="kw">then </span>
( <font color="Maroon" title="c4">f1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_3.html#T18" target="_self" onmouseover="rs('nfcont_3/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:28"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">X1</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="nfcont_3.html#T19" target="_self" onmouseover="rs('nfcont_3/T19')" onmouseout="rh()">Th19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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