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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T30" title="FUNCT_4:th.30">FUNCT_4:30</a></span>;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:27"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></span>;<br/>
<a NAME="E4:27"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></span>;<br/>
<span class="kw">thus </span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="relat_1.html#T12" title="RELAT_1:th.12">RELAT_1:12</a></span>
; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E6:27"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T23" target="_self" title="ALGSPEC1:th.23">Th24</a></span>;<br/>
<a NAME="E7:27"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#T23" target="_self" title="ALGSPEC1:th.23">Th24</a></span>;<br/>
<span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_4.html#T69" title="FUNCT_4:th.69">FUNCT_4:69</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algspec1.html#D2" target="_self" title="ALGSPEC1:def.2">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="relat_1.html#T12" title="RELAT_1:th.12">RELAT_1:12</a></span>
;<br/> <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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