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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</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">RNS</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">X</font>, <font color="Olive" title="b3">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Olive" title="b1">RNS</font>  st <font color="Olive" title="b4">f1</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b5">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b3">X1</font> holds <br/>( <font color="Olive" title="b4">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b5">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">X1</font> &amp; <font color="Olive" title="b4">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b5">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">X1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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">X</font>, <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>  st <font color="Olive" title="b3">f1</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b4">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">X1</font> holds <br/>( <font color="Olive" title="b3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b4">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <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> &amp; <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> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <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><br/><b>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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 <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>  st <font color="Olive" title="b1">f1</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Olive" title="b2">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X1</font> holds <br/>( <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> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> &amp; <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> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> )</span><br/><b>let </b><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f1</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X1</font> implies ( <font color="Maroon" title="c5">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> ) )</span><br/>









<b>assume </b><a NAME="E1:85"/>
( <font color="Maroon" title="c5">f1</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">X1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> )</span><br/>

<a NAME="E2:85"/><b>then </b>
( <font color="Maroon" title="c5">f1</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> &amp; <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> )
 
<b>by </b><i><a class="ref" href="ncfcont1.html#T78" target="_self" title="NCFCONT1:th.78">Th78</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E3:85"/>
( <font color="Maroon" title="c5">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> &amp; <font color="Maroon" title="c5">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c6">f2</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X1</font> )
 <b>by </b><i><a class="ref" href="ncfcont1.html#T84" target="_self" title="NCFCONT1:th.84">Th84</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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