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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font>, <font color="Olive" title="b6">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H</font> holds <br/>( <font color="Olive" title="b3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">G</font> iff for <font color="Olive" title="b7">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b6">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b7">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b7">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c4">H</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H</font> holds <br/>( <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font> iff for <font color="Olive" title="b4">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b3">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H</font> implies ( <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font> iff for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )</span><br/>













<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font> iff for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c10">h1</font> = <font color="Maroon" title="c4">H</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E3:35"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 <a href="c0sp1.html#K7" title="C0SP1:func.7">R_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of  <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T6" target="_self" title="C0SP1:th.6">Th6</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/><a NAME="E2:35_1"/>
<font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="c0sp1.html#T8" target="_self" title="C0SP1:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:35_1"/>
<font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="funcsdom.html#T10" title="FUNCSDOM:th.10">FUNCSDOM:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:35"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font></span><br/>

<a NAME="E6:35"/><span class="kw">then </span>
<font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="funcsdom.html#T10" title="FUNCSDOM:th.10">FUNCSDOM:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35"/>
<font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="c0sp1.html#T8" target="_self" title="C0SP1:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
