let X be set ; :: thesis: for r being Real
for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
r (#) f is_uniformly_continuous_on X
let r be Real; :: thesis: for RNS being RealNormSpace
for CNS being ComplexNormSpace
for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
r (#) f is_uniformly_continuous_on X
let RNS be RealNormSpace; :: thesis: for CNS being ComplexNormSpace
for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
r (#) f is_uniformly_continuous_on X
let CNS be ComplexNormSpace; :: thesis: for f being PartFunc of CNS,RNS st f is_uniformly_continuous_on X holds
r (#) f is_uniformly_continuous_on X
let f be PartFunc of CNS,RNS; :: thesis: ( f is_uniformly_continuous_on X implies r (#) f is_uniformly_continuous_on X )
assume A1: f is_uniformly_continuous_on X ; :: thesis: r (#) f is_uniformly_continuous_on X
then X c= dom f ;
hence A2: X c= dom (r (#) f) by VFUNCT_1:def 4; :: according to NCFCONT2:def 2 :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < r ) )

now :: thesis: for p being Real st 0 < p holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) )

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

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

let p be Real; :: thesis: ( 0 < p implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) ) )
assume A4: 0 < p ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) )
then consider s being Real such that
A5: 0 < s and
for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p by A1;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) )
thus 0 < s by A5; :: thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p
let x1, x2 be Point of CNS; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p )
assume that
A6: x1 in X and
A7: x2 in X and
||.(x1 - x2).|| < s ; :: thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A2, A6, VFUNCT_1:def 4
.= ||.((0. RNS) - ((r (#) f) /. x2)).|| by A3, RLVECT_1:10
.= ||.((0. RNS) - (r * (f /. x2))).|| by A2, A7, VFUNCT_1:def 4
.= ||.((0. RNS) - (0. RNS)).|| by A3, RLVECT_1:10
.= ||.(0. RNS).|| by RLVECT_1:13
.= 0 by NORMSP_0:def 6 ;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p by A4; :: thesis: verum

end;

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

let p be Real; :: thesis: ( 0 < p implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) ) )
A9: 0 < |.r.| by A8, COMPLEX1:47;
assume 0 < p ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) )
then 0 < p / |.r.| by A9;
then consider s being Real such that
A10: 0 < s and
A11: for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p / |.r.| by A1;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p ) )
thus 0 < s by A10; :: thesis: for x1, x2 being Point of CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p
let x1, x2 be Point of CNS; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p )
assume that
A12: x1 in X and
A13: x2 in X and
A14: ||.(x1 - x2).|| < s ; :: thesis: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p
A15: ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| = ||.((r * (f /. x1)) - ((r (#) f) /. x2)).|| by A2, A12, VFUNCT_1:def 4
.= ||.((r * (f /. x1)) - (r * (f /. x2))).|| by A2, A13, VFUNCT_1:def 4
.= ||.(r * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= |.r.| * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;
|.r.| * ||.((f /. x1) - (f /. x2)).|| < (p / |.r.|) * |.r.| by A9, A11, A12, A13, A14, XREAL_1:68;
hence ||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < p 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 CNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((r (#) f) /. x1) - ((r (#) f) /. x2)).|| < r ) ) ; :: thesis: verum