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

now :: thesis: for r1 being Element of REAL st r1 in rng (f | (left_open_halfline p)) holds
ex N being Neighbourhood of r1 st N c= rng (f | (left_open_halfline p))

let r1 be Element of REAL ; :: thesis: ( r1 in rng (f | (left_open_halfline p)) implies ex N being Neighbourhood of r1 st N c= rng (f | (left_open_halfline p)) )
set f1 = f | (left_open_halfline p);
assume r1 in rng (f | (left_open_halfline p)) ; :: thesis: ex N being Neighbourhood of r1 st N c= rng (f | (left_open_halfline p))
then consider x0 being Element of REAL such that
A4: x0 in dom (f | (left_open_halfline p)) and
A5: (f | (left_open_halfline p)) . x0 = r1 by PARTFUN1:3;
A6: r1 = f . x0 by A4, A5, FUNCT_1:47;
A7: x0 in (dom f) /\ (left_open_halfline p) by A4, RELAT_1:61;
then x0 in left_open_halfline p by XBOOLE_0:def 4;
then consider N being Neighbourhood of x0 such that
A8: N c= left_open_halfline p by RCOMP_1:18;
consider r being Real such that
A9: 0 < r and
A10: N = ].(x0 - r),(x0 + r).[ by RCOMP_1:def 6;
reconsider r = r as Element of REAL by XREAL_0:def 1;
0 < r / 2 by A9, XREAL_1:215;
then A11: x0 - (r / 2) < x0 - 0 by XREAL_1:15;
A12: r / 2 < r by A9, XREAL_1:216;
then A13: x0 - r < x0 - (r / 2) by XREAL_1:15;
A14: N c= dom f by A1, A8;
set fp = f . (x0 + (r / 2));
set fm = f . (x0 - (r / 2));
A15: x0 + (r / 2) < x0 + r by A12, XREAL_1:8;
A16: x0 < x0 + (r / 2) by A9, XREAL_1:29, XREAL_1:215;
then A17: x0 - (r / 2) < x0 + (r / 2) by A11, XXREAL_0:2;
x0 < x0 + r by A9, XREAL_1:29;
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 (left_open_halfline p) /\ (dom f) by A8, A10, A14, XBOOLE_0:def 4;
x0 - r < x0 by A9, XREAL_1:44;
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 (left_open_halfline p) /\ (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;
f | N is continuous by A2, A8, FCONT_1:16;
then A23: f | [.(x0 - (r / 2)),(x0 + (r / 2)).] is continuous by A10, A22, FCONT_1:16;
A24: [.(x0 - (r / 2)),(x0 + (r / 2)).] c= left_open_halfline p by A8, A10, A18, A20, XXREAL_2:def 12;

now :: thesis: ex N1 being Neighbourhood of r1 st N1 c= rng (f | (left_open_halfline p))

per cases ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) by A3;

suppose A25: f | (left_open_halfline p) is increasing ; :: thesis: ex N1 being Neighbourhood of r1 st N1 c= rng (f | (left_open_halfline p))

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_2:20;
then A26: 0 < (f . (x0 + (r / 2))) - (f . x0) by XREAL_1:50;
f . (x0 - (r / 2)) < f . x0 by A7, A11, A19, A25, RFUNCT_2:20;
then 0 < (f . x0) - (f . (x0 - (r / 2))) by XREAL_1:50;
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 6;
take N1 = N1; :: thesis: N1 c= rng (f | (left_open_halfline p))
f . (x0 - (r / 2)) < f . (x0 + (r / 2)) by A17, A21, A19, A25, RFUNCT_2:20;
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 | (left_open_halfline p)) :: thesis: verum

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N1 or x in rng (f | (left_open_halfline p)) )
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 | (left_open_halfline p))
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_1:7;
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_1:13;
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, A24, A23, A17, A27, A29, A28, FCONT_2:15, XBOOLE_1:1;
s in N by A10, A22, A33;
then s in (dom f) /\ (left_open_halfline p) by A8, A14, XBOOLE_0:def 4;
then A35: s in dom (f | (left_open_halfline p)) by RELAT_1:61;
then x = (f | (left_open_halfline p)) . s by A34, FUNCT_1:47;
hence x in rng (f | (left_open_halfline p)) by A35, FUNCT_1:def 3; :: thesis: verum

end;

end;

suppose A36: f | (left_open_halfline p) is decreasing ; :: thesis: ex N1 being Neighbourhood of r1 st N1 c= rng (f | (left_open_halfline p))

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_2:21;
then A37: 0 < (f . x0) - (f . (x0 + (r / 2))) by XREAL_1:50;
f . x0 < f . (x0 - (r / 2)) by A7, A11, A19, A36, RFUNCT_2:21;
then 0 < (f . (x0 - (r / 2))) - (f . x0) by XREAL_1:50;
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 6;
take N1 = N1; :: thesis: N1 c= rng (f | (left_open_halfline p))
f . (x0 + (r / 2)) < f . (x0 - (r / 2)) by A17, A21, A19, A36, RFUNCT_2:21;
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 | (left_open_halfline p)) :: thesis: verum

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N1 or x in rng (f | (left_open_halfline p)) )
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 | (left_open_halfline p))
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_1:7;
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_1:13;
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, A24, A23, A17, A38, A40, A39, FCONT_2:15, XBOOLE_1:1;
s in N by A10, A22, A44;
then s in (dom f) /\ (left_open_halfline p) by A8, A14, XBOOLE_0:def 4;
then A46: s in dom (f | (left_open_halfline p)) by RELAT_1:61;
then x = (f | (left_open_halfline p)) . s by A45, FUNCT_1:47;
hence x in rng (f | (left_open_halfline p)) by A46, FUNCT_1:def 3; :: thesis: verum

end;

end;

end;

end;

hence ex N being Neighbourhood of r1 st N c= rng (f | (left_open_halfline p)) ; :: thesis: verum

end;

hence rng (f | (left_open_halfline p)) is open by RCOMP_1:20; :: thesis: verum