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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies for <font color="Olive" title="b1">lp</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="normsp_1.html#L1" title="NORMSP_1:struct.1">NORMSTR</a>   st <font color="Olive" title="b1">lp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" title="LP_SPACE:func.3">l_norm^</a> <font color="Maroon" title="c1">p</font></span>)</span> #) holds <br/><font color="Olive" title="b1">lp</font> is <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a> )</span><br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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">lp</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="normsp_1.html#L1" title="NORMSP_1:struct.1">NORMSTR</a>   st <font color="Olive" title="b1">lp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" title="LP_SPACE:func.3">l_norm^</a> <font color="Maroon" title="c1">p</font></span>)</span> #) holds <br/><font color="Olive" title="b1">lp</font> is <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a></span><br/>

<b>let </b><font color="Maroon" title="c2">lp</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="normsp_1.html#L1" title="NORMSP_1:struct.1">NORMSTR</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">lp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" title="LP_SPACE:func.3">l_norm^</a> <font color="Maroon" title="c1">p</font></span>)</span> #) implies <font color="Maroon" title="c2">lp</font> is <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a> )</span><br/>

<b>assume </b><a NAME="E2:23"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">lp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" title="LP_SPACE:func.3">l_norm^</a> <font color="Maroon" title="c1">p</font></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="c2">lp</font> is <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a></span><br/>

<a NAME="E3:23"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">lp</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( ( <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1: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>  implies <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> <font color="Maroon" title="c2">lp</font> ) &amp; ( <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> <font color="Maroon" title="c2">lp</font> implies <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1: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>  ) &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> )
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="#E2:23"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="lp_space.html#T13" target="_self" title="LP_SPACE:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:23"/>
<font color="Maroon" title="c2">lp</font> is <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_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/>


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