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<a NAME="E1:69_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="c0sp2.html#K7" title="C0SP2:func.7">R_VectorSpace_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>
 
;<br/>
<a NAME="E2:69_1_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 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="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp2.html#T34" target="_self" title="C0SP2:th.34">Th34</a></span>;<br/>
<a NAME="E3:69_1_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp2.html#T34" target="_self" title="C0SP2:th.34">Th34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:69_1_1"/>
(  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="c0sp2.html#K9" title="C0SP2:func.9">R_Normed_Space_of_C_0_Functions</a> <font color="Maroon" title="c1">X</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:69_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="c0sp2.html#T35" target="_self" title="C0SP2:th.35">Th35</a>, <a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</a>, <a class="ref" href="normsp_0.html#D6" title="NORMSP_0:def.6">NORMSP_0:def 6</a>, <a class="ref" href="rsspace3.html#T2" title="RSSPACE3:th.2">RSSPACE3:2</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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