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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</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">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b3">r0</font>, <font color="Olive" title="b4">p0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p0</font> holds <br/>( <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r0</font> iff  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b3">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <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">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">r0</font>, <font color="Olive" title="b3">p0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p0</font> holds <br/>( <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r0</font> iff  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b2">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">r0</font>, <font color="Olive" title="b2">p0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p0</font> holds <br/>( <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r0</font> iff  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">r0</font>, <font color="Maroon" title="c5">p0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span>; <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="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p0</font> implies ( <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font> iff  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c4">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E16">AS0</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p0</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="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font> iff  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c4">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )</span><br/>

<div><span class="kw">hereby </span> <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">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c4">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E17">AS1</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</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">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c4">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</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">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">i0</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="ndiff_5.html#E10"><span class="lab"><font color="Green" title="E10">ZE</font></span></a></span>;<br/><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E18">X1</font></span>: 
<span class="p1">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c7">i0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D8" title="PRVECT_2:def.8">PRVECT_2:def 8</a></span>;<br/><a NAME="E3:20_1_1"/>
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>,<span class="p1"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K13" title="PRVECT_2:func.13">productnorm</a> <font color="Maroon" title="c1">G</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" title="PRVECT_2:th.6">PRVECT_2:6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:20_1_1"/>
<font color="Maroon" title="c4">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">AS0</font></span></a>, <a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E17">AS1</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E18">X1</font></span></a>, <a class="ref" href="prvect_2.html#D2" title="PRVECT_2:def.2">PRVECT_2: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/></div>
<span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:20"/><span class="lab"><font color="Green" title="E17">AS2</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c4">r0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ; <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="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">rp</font> = <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T10" target="_self" title="NDIFF_5:th.10">LM001</a></span>;<br/>
<a NAME="E5:20"/><span class="lab"><font color="Green" title="E18">X1</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c6">rp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<a NAME="E6:20"/><span class="lab"><font color="Green" title="E19">X2</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c4">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i0</font> be    <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"> ( <font color="Maroon" title="c7">i0</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> <font color="Maroon" title="c6">rp</font> implies <font color="Maroon" title="c6">rp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> )</span><br/><span class="kw">assume </span><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E20">AS1</font></span>: 
<font color="Maroon" title="c7">i0</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> <font color="Maroon" title="c6">rp</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">rp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">i1</font> = <font color="Maroon" title="c7">i0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="ndiff_5.html#E10"><span class="lab"><font color="Green" title="E10">ZE</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E18">X1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c7">i0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E20">AS1</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E18">X1</font></span></a></span>;<br/><a NAME="E4:20_2"/><span class="lab"><font color="Green" title="E21">P1</font></span>: 
 <a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="prvect_2.html#K4" title="PRVECT_2:func.4">carr</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K8" title="PRVECT_2:func.8">zeros</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">addop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>,<span class="p1"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K13" title="PRVECT_2:func.13">productnorm</a> <font color="Maroon" title="c1">G</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#T6" title="PRVECT_2:th.6">PRVECT_2:6</a></span>;<br/><a NAME="E5:20_2"/>
<span class="p1">(<span class="default"><a href="prvect_2.html#K9" title="PRVECT_2:func.9">multop</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_2.html#K1" title="PRVECT_2:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K5" title="PRVECT_2:func.5">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_2.html#D8" title="PRVECT_2:def.8">PRVECT_2:def 8</a></span>;<br/><a NAME="E6:20_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">rp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p0</font> <a href="ndiff_5.html#K1" title="NDIFF_5:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E16">AS0</font></span></a>, <a class="txt" href="#E4:20_2"><span class="lab"><font color="Green" title="E21">P1</font></span></a>, <a class="ref" href="prvect_2.html#D2" title="PRVECT_2:def.2">PRVECT_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:20_2"/>
<font color="Maroon" title="c6">rp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">AS2</font></span></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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:20"/>
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E18">X1</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E19">X2</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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/>


</div>
