let p, g be Real; :: thesis: for f being PartFunc of REAL ,REAL st ].p,g.[ c= dom f & f | ].p,g.[ is continuous & ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) holds
rng (f | ].p,g.[) is open
let f be PartFunc of REAL ,REAL ; :: thesis: ( ].p,g.[ c= dom f & f | ].p,g.[ is continuous & ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) implies rng (f | ].p,g.[) is open )
assume that
A1: ].p,g.[ c= dom f and
A2: f | ].p,g.[ is continuous and
A3: ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) ; :: thesis: rng (f | ].p,g.[) is open

now

let r1 be real number ; :: thesis: ( r1 in rng (f | ].p,g.[) implies ex N being Neighbourhood of r1 st N c= rng (f | ].p,g.[) )
set f1 = f | ].p,g.[;
assume r1 in rng (f | ].p,g.[) ; :: thesis: ex N being Neighbourhood of r1 st N c= rng (f | ].p,g.[)
then consider x0 being Real such that
A4: x0 in dom (f | ].p,g.[) and
A5: (f | ].p,g.[) . x0 = r1 by PARTFUN1:26;
A6: r1 = f . x0 by A4, A5, FUNCT_1:70;
A7: x0 in (dom f) /\ ].p,g.[ by A4, RELAT_1:90;
then x0 in ].p,g.[ by XBOOLE_0:def 4;
then consider N being Neighbourhood of x0 such that
A8: N c= ].p,g.[ by RCOMP_1:39;
consider r being real number such that
A9: 0 < r and
A10: N = ].(x0 - r),(x0 + r).[ by RCOMP_1:def 7;
reconsider r = r as Real by XREAL_0:def 1;
0 < r / 2 by A9, XREAL_1:217;
then A11: x0 - (r / 2) < x0 - 0 by XREAL_1:17;
A12: r / 2 < r by A9, XREAL_1:218;
then A13: x0 - r < x0 - (r / 2) by XREAL_1:17;
A14: N c= dom f by A1, A8, XBOOLE_1:1;
set fp = f . (x0 + (r / 2));
set fm = f . (x0 - (r / 2));
A15: x0 + (r / 2) < x0 + r by A12, XREAL_1:10;
A16: x0 < x0 + (r / 2) by A9, XREAL_1:31, XREAL_1:217;
then A17: x0 - (r / 2) < x0 + (r / 2) by A11, XXREAL_0:2;
x0 < x0 + r by A9, XREAL_1:31;
then x0 - (r / 2) < x0 + r by A11, XXREAL_0:2;
then A18: x0 - (r / 2) in ].(x0 - r),(x0 + r).[ by A13;
then A19: x0 - (r / 2) in ].p,g.[ /\ (dom f) by A8, A10, A14, XBOOLE_0:def 4;
x0 - r < x0 by A9, XREAL_1:46;
then x0 - r < x0 + (r / 2) by A16, XXREAL_0:2;
then A20: x0 + (r / 2) in ].(x0 - r),(x0 + r).[ by A15;
then A21: x0 + (r / 2) in ].p,g.[ /\ (dom f) by A8, A10, A14, XBOOLE_0:def 4;
A22: [.(x0 - (r / 2)),(x0 + (r / 2)).] c= ].(x0 - r),(x0 + r).[ by A18, A20, XXREAL_2:def 12;
A23: [.(x0 - (r / 2)),(x0 + (r / 2)).] c= ].p,g.[ by A8, A10, A18, A20, XXREAL_2:def 12;
then A24: f | [.(x0 - (r / 2)),(x0 + (r / 2)).] is continuous by A2, FCONT_1:17;

now

per cases ( f | ].p,g.[ is increasing or f | ].p,g.[ is decreasing ) by A3;

suppose A25: f | ].p,g.[ is increasing ; :: thesis: ex N1 being Neighbourhood of r1 st N1 c= rng (f | ].p,g.[)

set R = min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0));
f . x0 < f . (x0 + (r / 2)) by A7, A16, A21, A25, RFUNCT_2:43;
then A26: 0 < (f . (x0 + (r / 2))) - (f . x0) by XREAL_1:52;
f . (x0 - (r / 2)) < f . x0 by A7, A11, A19, A25, RFUNCT_2:43;
then 0 < (f . x0) - (f . (x0 - (r / 2))) by XREAL_1:52;
then 0 < min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0)) by A26, XXREAL_0:15;
then reconsider N1 = ].(r1 - (min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0)))),(r1 + (min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0)))).[ as Neighbourhood of r1 by RCOMP_1:def 7;
take N1 = N1; :: thesis: N1 c= rng (f | ].p,g.[)
f . (x0 - (r / 2)) < f . (x0 + (r / 2)) by A17, A21, A19, A25, RFUNCT_2:43;
then [.(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).] = {} by XXREAL_1:29;
then A27: [.(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).] \/ [.(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).] = [.(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).] ;
thus N1 c= rng (f | ].p,g.[) :: thesis: verum

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in N1 or x in rng (f | ].p,g.[) )
A28: ].(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).[ c= [.(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).] by XXREAL_1:25;
assume x in N1 ; :: thesis: x in rng (f | ].p,g.[)
then consider r2 being Real such that
A29: r2 = x and
A30: (f . x0) - (min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0))) < r2 and
A31: r2 < (f . x0) + (min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0))) by A6;
min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0)) <= (f . (x0 + (r / 2))) - (f . x0) by XXREAL_0:17;
then (f . x0) + (min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0))) <= (f . x0) + ((f . (x0 + (r / 2))) - (f . x0)) by XREAL_1:9;
then A32: r2 < f . (x0 + (r / 2)) by A31, XXREAL_0:2;
min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0)) <= (f . x0) - (f . (x0 - (r / 2))) by XXREAL_0:17;
then (f . x0) - ((f . x0) - (f . (x0 - (r / 2)))) <= (f . x0) - (min ((f . x0) - (f . (x0 - (r / 2)))),((f . (x0 + (r / 2))) - (f . x0))) by XREAL_1:15;
then f . (x0 - (r / 2)) < r2 by A30, XXREAL_0:2;
then r2 in ].(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).[ by A32;
then consider s being Real such that
A33: s in [.(x0 - (r / 2)),(x0 + (r / 2)).] and
A34: x = f . s by A1, A23, A24, A17, A27, A29, A28, FCONT_2:16, XBOOLE_1:1;
s in N by A10, A22, A33;
then s in (dom f) /\ ].p,g.[ by A8, A14, XBOOLE_0:def 4;
then A35: s in dom (f | ].p,g.[) by RELAT_1:90;
then x = (f | ].p,g.[) . s by A34, FUNCT_1:70;
hence x in rng (f | ].p,g.[) by A35, FUNCT_1:def 5; :: thesis: verum

end;

end;

suppose A36: f | ].p,g.[ is decreasing ; :: thesis: ex N1 being Neighbourhood of r1 st N1 c= rng (f | ].p,g.[)

set R = min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2))));
f . (x0 + (r / 2)) < f . x0 by A7, A16, A21, A36, RFUNCT_2:44;
then A37: 0 < (f . x0) - (f . (x0 + (r / 2))) by XREAL_1:52;
f . x0 < f . (x0 - (r / 2)) by A7, A11, A19, A36, RFUNCT_2:44;
then 0 < (f . (x0 - (r / 2))) - (f . x0) by XREAL_1:52;
then 0 < min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2)))) by A37, XXREAL_0:15;
then reconsider N1 = ].(r1 - (min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2)))))),(r1 + (min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2)))))).[ as Neighbourhood of r1 by RCOMP_1:def 7;
take N1 = N1; :: thesis: N1 c= rng (f | ].p,g.[)
f . (x0 + (r / 2)) < f . (x0 - (r / 2)) by A17, A21, A19, A36, RFUNCT_2:44;
then [.(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).] = {} by XXREAL_1:29;
then A38: [.(f . (x0 - (r / 2))),(f . (x0 + (r / 2))).] \/ [.(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).] = [.(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).] ;
thus N1 c= rng (f | ].p,g.[) :: thesis: verum

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in N1 or x in rng (f | ].p,g.[) )
A39: ].(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).[ c= [.(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).] by XXREAL_1:25;
assume x in N1 ; :: thesis: x in rng (f | ].p,g.[)
then consider r2 being Real such that
A40: r2 = x and
A41: (f . x0) - (min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2))))) < r2 and
A42: r2 < (f . x0) + (min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2))))) by A6;
min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2)))) <= (f . (x0 - (r / 2))) - (f . x0) by XXREAL_0:17;
then (f . x0) + (min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2))))) <= (f . x0) + ((f . (x0 - (r / 2))) - (f . x0)) by XREAL_1:9;
then A43: r2 < f . (x0 - (r / 2)) by A42, XXREAL_0:2;
min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2)))) <= (f . x0) - (f . (x0 + (r / 2))) by XXREAL_0:17;
then (f . x0) - ((f . x0) - (f . (x0 + (r / 2)))) <= (f . x0) - (min ((f . (x0 - (r / 2))) - (f . x0)),((f . x0) - (f . (x0 + (r / 2))))) by XREAL_1:15;
then f . (x0 + (r / 2)) < r2 by A41, XXREAL_0:2;
then r2 in ].(f . (x0 + (r / 2))),(f . (x0 - (r / 2))).[ by A43;
then consider s being Real such that
A44: s in [.(x0 - (r / 2)),(x0 + (r / 2)).] and
A45: x = f . s by A1, A23, A24, A17, A38, A40, A39, FCONT_2:16, XBOOLE_1:1;
s in N by A10, A22, A44;
then s in (dom f) /\ ].p,g.[ by A8, A14, XBOOLE_0:def 4;
then A46: s in dom (f | ].p,g.[) by RELAT_1:90;
then x = (f | ].p,g.[) . s by A45, FUNCT_1:70;
hence x in rng (f | ].p,g.[) by A46, FUNCT_1:def 5; :: thesis: verum

end;

end;

end;

end;

hence ex N being Neighbourhood of r1 st N c= rng (f | ].p,g.[) ; :: thesis: verum

end;

hence rng (f | ].p,g.[) is open by RCOMP_1:41; :: thesis: verum