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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">r2</font> holds <br/><font color="Olive" title="b1">p1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Olive" title="b4">r2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r2</font> holds <br/><font color="Maroon" title="c2">p1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Olive" title="b2">r2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">r1</font>, <font color="Maroon" title="c5">r2</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r2</font> implies <font color="Maroon" title="c2">p1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c5">r2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r2</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="c2">p1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c5">r2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">s1</font> = <font color="Maroon" title="c4">r1</font>, <font color="Maroon" title="c7">s2</font> = <font color="Maroon" title="c5">r2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="euclid.html#T22" title="EUCLID:th.22">EUCLID:22</a></span>;<br/>
<a NAME="E3:1"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E3">XX</font></span>:  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">RLSStruct</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K8" title="FUNCSDOM:func.8">RealFuncZero</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funcsdom.html#K7" title="FUNCSDOM:func.7">RealFuncExtMult</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#D6" title="FUNCSDOM:def.6">FUNCSDOM:def 6</a></span>
<br/>.= 
 <a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>

;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">s2</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T93" title="FINSEQ_2:th.93">FINSEQ_2:93</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">p1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">p2</font> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K5" title="FUNCSDOM:func.5">RealFuncAdd</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">r1</font>,<font color="Maroon" title="c5">r2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E3">XX</font></span></a>, <a class="ref" href="algstr_0.html#D1" title="ALGSTR_0:def.1">ALGSTR_0:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> <a href="funcsdom.html#K3" title="FUNCSDOM:func.3">.:</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#D1" title="FUNCSDOM:def.1">FUNCSDOM:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">s1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c7">s2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D4" title="RVSUM_1:def.4">RVSUM_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c4">r1</font> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <font color="Maroon" title="c5">r2</font>

; <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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