let X be set ; :: thesis: for z being Complex
for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of RNS,CNS st f is_uniformly_continuous_on X holds
z (#) f is_uniformly_continuous_on X
let z be Complex; :: thesis: for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of RNS,CNS st f is_uniformly_continuous_on X holds
z (#) f is_uniformly_continuous_on X
let RNS be RealNormSpace; :: thesis: for CNS being ComplexNormSpace
for f being PartFunc of RNS,CNS st f is_uniformly_continuous_on X holds
z (#) f is_uniformly_continuous_on X
let CNS be ComplexNormSpace; :: thesis: for f being PartFunc of RNS,CNS st f is_uniformly_continuous_on X holds
z (#) f is_uniformly_continuous_on X
let f be PartFunc of RNS,CNS; :: thesis: ( f is_uniformly_continuous_on X implies z (#) f is_uniformly_continuous_on X )
assume A1: f is_uniformly_continuous_on X ; :: thesis: z (#) f is_uniformly_continuous_on X
then X c= dom f by Def3;
hence A2: X c= dom (z (#) f) by VFUNCT_2:def 2; :: according to NCFCONT2:def 3 :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )

now

per cases ( z = 0 or z <> 0 ) ;

suppose A3: z = 0 ; :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )

let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) ) )
assume A4: 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )
then consider s being Real such that
A5: 0 < s and
for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r by A1, Def3;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )
thus 0 < s by A5; :: thesis: for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r
let x1, x2 be Point of RNS; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r )
assume that
A6: x1 in X and
A7: x2 in X and
||.(x1 - x2).|| < s ; :: thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A2, A6, VFUNCT_2:def 2
.= ||.((0. CNS) - ((z (#) f) /. x2)).|| by A3, CLVECT_1:1
.= ||.((0. CNS) - (z * (f /. x2))).|| by A2, A7, VFUNCT_2:def 2
.= ||.((0. CNS) - (0. CNS)).|| by A3, CLVECT_1:1
.= ||.(0. CNS).|| by RLVECT_1:13
.= 0 by NORMSP_0:def 6 ;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r by A4; :: thesis: verum

end;

suppose A8: z <> 0 ; :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )

let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) ) )
A9: 0 < |.z.| by A8, COMPLEX1:47;
assume 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )
then 0 < r / |.z.| by A9, XREAL_1:139;
then consider s being Real such that
A10: 0 < s and
A11: for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r / |.z.| by A1, Def3;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) )
thus 0 < s by A10; :: thesis: for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r
let x1, x2 be Point of RNS; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r )
assume that
A12: x1 in X and
A13: x2 in X and
A14: ||.(x1 - x2).|| < s ; :: thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r
A15: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A2, A12, VFUNCT_2:def 2
.= ||.((z * (f /. x1)) - (z * (f /. x2))).|| by A2, A13, VFUNCT_2:def 2
.= ||.(z * ((f /. x1) - (f /. x2))).|| by CLVECT_1:9
.= |.z.| * ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:def 13 ;
|.z.| * ||.((f /. x1) - (f /. x2)).|| < (r / |.z.|) * |.z.| by A9, A11, A12, A13, A14, XREAL_1:68;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r by A9, A15, XCMPLX_1:87; :: thesis: verum

end;

end;

end;

hence for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r ) ) ; :: thesis: verum