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 4; :: 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 & ( 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 A6: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s ) ; :: thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r
then ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A2, VFUNCT_2:def 4
.= ||.((0. CNS) - ((z (#) f) /. x2)).|| by A3, CLVECT_1:2
.= ||.((0. CNS) - (z * (f /. x2))).|| by A2, A6, VFUNCT_2:def 4
.= ||.((0. CNS) - (0. CNS)).|| by A3, CLVECT_1:2
.= ||.(0. CNS).|| by RLVECT_1:26
.= 0 by CLVECT_1:def 11 ;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r by A4; :: thesis: verum

end;

suppose 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 ) )

then A7: ( 0 < |.z.| & 0 <> |.z.| ) by COMPLEX1:133;
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 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 A7, XREAL_1:141;
then consider s being Real such that
A8: ( 0 < s & ( 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 A8; :: 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 A9: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s ) ; :: thesis: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r
then A10: ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| = ||.((z * (f /. x1)) - ((z (#) f) /. x2)).|| by A2, VFUNCT_2:def 4
.= ||.((z * (f /. x1)) - (z * (f /. x2))).|| by A2, A9, VFUNCT_2:def 4
.= ||.(z * ((f /. x1) - (f /. x2))).|| by CLVECT_1:10
.= |.z.| * ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:def 11 ;
|.z.| * ||.((f /. x1) - (f /. x2)).|| < (r / |.z.|) * |.z.| by A7, A8, A9, XREAL_1:70;
hence ||.(((z (#) f) /. x1) - ((z (#) f) /. x2)).|| < r by A7, A10, XCMPLX_1:88; :: 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