let X be non empty TopSpace; :: thesis: for f1 being Function of X,R^1
for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
let f1 be Function of X,R^1; :: thesis: for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
let a be real number ; :: thesis: ( f1 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous ) )
defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds
$2 = a * r1;
A1: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; :: thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
reconsider r3 = a * r1 as Element of REAL ;
for r1 being Real st f1 . x = r1 holds
r3 = a * r1 ;
hence ex y being Element of REAL st
for r1 being Real st f1 . x = r1 holds
y = a * r1 ; :: thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch 3(A1);
 then consider f being Function of the carrier of X,REAL such that
A2: for x being Element of X
for r1 being Real st f1 . x = r1 holds
f . x = a * r1 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
A3: for p being Point of X
for r1 being real number st f1 . p = r1 holds
g0 . p = a * r1
proof
let p be Point of X; :: thesis: for r1 being real number st f1 . p = r1 holds
g0 . p = a * r1
let r1 be real number ; :: thesis: ( f1 . p = r1 implies g0 . p = a * r1 )
assume A4: f1 . p = r1 ; :: thesis: g0 . p = a * r1
reconsider r1 = r1 as Element of REAL by XREAL_0:def 1;
g0 . p = a * r1 by A2, A4;
hence g0 . p = a * r1 ; :: thesis: verum
end;
assume A5: f1 is continuous ; :: thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; :: thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; :: thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; :: thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A6: r0 > 0 and
A7: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
A8: r = a * r1 by A2;
A9: r = a * r1 by A2;
now
per cases ( a >= 0 or a < 0 ) ;
case A10: a >= 0 ; :: thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
now
per cases ( a > 0 or a = 0 ) by A10;
case A11: a > 0 ; :: thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = r0 / a;
reconsider G1 = ].(r1 - (r0 / a)),(r1 + (r0 / a)).[ as Subset of R^1 by TOPMETR:17;
A12: r1 < r1 + (r0 / a) by A6, A11, XREAL_1:29, XREAL_1:139;
then r1 - (r0 / a) < r1 by XREAL_1:19;
then A13: f1 . p in G1 by A12, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A14: ( p in W1 & W1 is open ) and
A15: f1 .: W1 c= G1 by A5, A13, Th20;
set W = W1;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; :: thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A16: z in dom g0 and
A17: z in W1 and
A18: g0 . z = x by FUNCT_1:def 6;
reconsider pz = z as Point of X by A16;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A19: x = a * aa1 by A2, A18;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def 1;
then A20: f1 . pz in f1 .: W1 by A17, FUNCT_1:def 6;
then r1 - (r0 / a) < aa1 by A15, XXREAL_1:4;
then A21: a * (r1 - (r0 / a)) < a * aa1 by A11, XREAL_1:68;
reconsider rx = x as Real by A18, XREAL_0:def 1;
A22: a * (r1 + (r0 / a)) = (a * r1) + (a * (r0 / a))
.= r + r0 by A8, A11, XCMPLX_1:87 ;
A23: a * (r1 - (r0 / a)) = (a * r1) - (a * (r0 / a))
.= r - r0 by A8, A11, XCMPLX_1:87 ;
aa1 < r1 + (r0 / a) by A15, A20, XXREAL_1:4;
then rx < r + r0 by A11, A19, A22, XREAL_1:68;
hence x in ].(r - r0),(r + r0).[ by A19, A21, A23, XXREAL_1:4; :: thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A7, A14, XBOOLE_1:1; :: thesis: verum
end;
case A24: a = 0 ; :: thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = r0;
reconsider G1 = ].(r1 - r0),(r1 + r0).[ as Subset of R^1 by TOPMETR:17;
A25: r1 < r1 + r0 by A6, XREAL_1:29;
then r1 - r0 < r1 by XREAL_1:19;
then A26: f1 . p in G1 by A25, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A27: ( p in W1 & W1 is open ) and
f1 .: W1 c= G1 by A5, A26, Th20;
set W = W1;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; :: thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A28: z in dom g0 and
z in W1 and
A29: g0 . z = x by FUNCT_1:def 6;
reconsider pz = z as Point of X by A28;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
x = a * aa1 by A2, A29
.= 0 by A24 ;
hence x in ].(r - r0),(r + r0).[ by A6, A9, A24, XXREAL_1:4; :: thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A7, A27, XBOOLE_1:1; :: thesis: verum
end;
end;
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; :: thesis: verum
end;
case A30: a < 0 ; :: thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = r0 / (- a);
reconsider G1 = ].(r1 - (r0 / (- a))),(r1 + (r0 / (- a))).[ as Subset of R^1 by TOPMETR:17;
- a > 0 by A30, XREAL_1:58;
then A31: r1 < r1 + (r0 / (- a)) by A6, XREAL_1:29, XREAL_1:139;
then r1 - (r0 / (- a)) < r1 by XREAL_1:19;
then A32: f1 . p in G1 by A31, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A33: ( p in W1 & W1 is open ) and
A34: f1 .: W1 c= G1 by A5, A32, Th20;
set W = W1;
- a <> 0 by A30;
then A35: (- a) * (r0 / (- a)) = r0 by XCMPLX_1:87;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; :: thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A36: z in dom g0 and
A37: z in W1 and
A38: g0 . z = x by FUNCT_1:def 6;
reconsider pz = z as Point of X by A36;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def 1;
then A39: f1 . pz in f1 .: W1 by A37, FUNCT_1:def 6;
then r1 - (r0 / (- a)) < aa1 by A34, XXREAL_1:4;
then A40: a * aa1 < a * (r1 - (r0 / (- a))) by A30, XREAL_1:69;
A41: a * (r1 + (r0 / (- a))) = (a * r1) - (- (a * (r0 / (- a))))
.= r - r0 by A3, A35 ;
A42: a * (r1 - (r0 / (- a))) = (a * r1) + (- (a * (r0 / (- a))))
.= r + r0 by A3, A35 ;
aa1 < r1 + (r0 / (- a)) by A34, A39, XXREAL_1:4;
then A43: r - r0 < a * aa1 by A30, A41, XREAL_1:69;
x = a * aa1 by A2, A38;
hence x in ].(r - r0),(r + r0).[ by A40, A42, A43, XXREAL_1:4; :: thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A7, A33, XBOOLE_1:1; :: thesis: verum
end;
end;
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; :: thesis: verum
end;
then g0 is continuous by Th20;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous ) by A3; :: thesis: verum