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<div class="add">

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V2</font>, <font color="Maroon" title="c3">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">V2</font>;<br/>









<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c4">f1</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> &amp; <font color="Maroon" title="c5">f2</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font>;<br/>

<b>assume </b><a NAME="E2:36"/><i><font color="Green" title="E31">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/>

<b>assume </b><a NAME="E3:36"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">b</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">b1</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">V1</font> <b>by </b><i><a class="ref" href="matrlin.html#D4" target="_self" title="MATRLIN:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>;<br/><a NAME="E1:36_1"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G4" title="VECTSP_1:aggr.4">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">V1</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c3">V1</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">V1</font> #)
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><a NAME="E2:36_1"/><b>then </b>
<font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T3" title="RLVECT_1:th.3">RLVECT_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">KL</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c7">b</font><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c9">KL</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12" title="VECTSP_7:th.12">VECTSP_7:12</a></i>;<br/><a NAME="E5:36_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c6">b1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="matrlin.html#D4" target="_self" title="MATRLIN:def.4">Def4</a></i>;<br/><a NAME="E6:36_1"/><i><font color="Green" title="E35">A6</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c9">KL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/><a NAME="E7:36_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:36"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">D</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c11">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">D</font> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font>;<br/><a NAME="E9:36_1"/><i><font color="Green" title="E36">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">D</font> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:36_1_1"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:36_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:36_1_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E3:36_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:36_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E3:36_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="matrlin.html#E4:36_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="vectsp_6.html#D8" title="VECTSP_6:def.8">VECTSP_6:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">D</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="matrlin.html#E3:36_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="matrlin.html#E7:36_1_1"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:36_1"/><i><font color="Green" title="E37">A11</font></i>: 
for <font color="Olive" title="b1">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">D</font> : verum </span>}</span>  holds <br/><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>;<br/>

<b>assume </b><a NAME="E1:36_1_2"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">D</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">D</font><b> such that </b><br/><a NAME="E3:36_1_2"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>
 ;<br/>
<font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
<font color="Maroon" title="c4">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:36"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<b>then </b><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:36"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="mod_2.html#D5" title="MOD_2:def.5">MOD_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:36"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="mod_2.html#D5" title="MOD_2:def.5">MOD_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E6:36_1_2"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">w</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:36_1_2"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font> = 
<font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E4:36_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="matrlin.html#E5:36_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="matrlin.html#E6:36_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="matrlin.html#T24" target="_self" title="MATRLIN:th.24">Th24</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">KL</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:36"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="matrlin.html#E9:36_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="matrlin.html#E10:36_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="matrlin.html#T25" target="_self" title="MATRLIN:th.25">Th25</a></i>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font>
<b>by </b><i><a class="txt" href="matrlin.html#E4:36_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="matrlin.html#E5:36_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="matrlin.html#E6:36_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="matrlin.html#T24" target="_self" title="MATRLIN:th.24">Th24</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:36"/>
<font color="Maroon" title="c4">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
