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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">f1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c2">f2</font> implies for <font color="Olive" title="b1">g1</font> being   <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font><br/> for <font color="Olive" title="b2">g2</font> being   <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c2">f2</font> holds  <font color="Olive" title="b1">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b2">g2</font> is    <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font> )</span><br/>

<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c2">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g1</font> being   <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font><br/> for <font color="Olive" title="b2">g2</font> being   <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c2">f2</font> holds  <font color="Olive" title="b1">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b2">g2</font> is    <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span><br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">f2</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T31" title="FUNCT_4:th.31">FUNCT_4:31</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">g1</font> be    <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g2</font> being   <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c2">f2</font> holds  <font color="Maroon" title="c3">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">g2</font> is    <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span><br/><b>let </b><font color="Maroon" title="c4">g2</font> be    <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c2">f2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">g2</font> is    <a href="algspec1.html#M1" title="ALGSPEC1:mode.1">rng-retract</a> of <font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span><br/>



<a NAME="E3:28"/><i><font color="Green" title="E26">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f2</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></i>;<br/>
<b>thus </b> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:28"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="algspec1.html#E3:28"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">ALGSPEC1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span></span><br/>

<a NAME="E5:28"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f2</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T24" target="_self" title="ALGSPEC1:th.24">Th24</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="funct_4.html#T73" title="FUNCT_4:th.73">FUNCT_4:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f1</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g2</font></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f1</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T23" title="FUNCT_4:th.23">FUNCT_4:23</a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E2:28"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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