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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b3">L</font> being    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b3">L</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a>  &amp;  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v2</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</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">L</font> being    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">L</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a>  &amp;  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v2</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">L</font> be    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of <font color="Maroon" title="c1">V</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">L</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a>  &amp;  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v2</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c4">L</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a> 
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font>
 ; <a class="txt" 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">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v2</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">L</font> = <font color="Maroon" title="c4">L</font> as    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E6:40"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
( <font color="Maroon" title="c6">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c5">L</font> )
 <span class="kw">and </span><br/><a NAME="E7:40"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 ex <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> &amp;  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</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> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="convex1.html#D3" target="_self" title="CONVEX1:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E9:40"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E10:40"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E11:40"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</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> <font color="Maroon" title="c7">f</font> holds <br/>( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<a NAME="E12:40"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E13:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>;<br/>
<a NAME="E14:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E16:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<a NAME="E17:40"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">r1</font> = <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E19:40"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E20:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<a NAME="E21:40"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">r2</font> = <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E23:40"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1"/>
( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> or <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c2">v1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ; <a class="txt" 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="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><div class="add"><a NAME="E2:40_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="rvsum_1.html#T77" title="RVSUM_1:th.77">RVSUM_1:77</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c2">v1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ; <a class="txt" 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="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><div class="add"><a NAME="E2:40_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E19:40"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E20:40"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E23:40"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="rvsum_1.html#T77" title="RVSUM_1:th.77">RVSUM_1:77</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></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E25:40"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="rlvect_2.html#T33" title="RLVECT_2:th.33">RLVECT_2:33</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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