let X be set ; :: thesis: for p being Real
for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
p (#) f is_uniformly_continuous_on X
let p be Real; :: thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
p (#) f is_uniformly_continuous_on X
let S, T be RealNormSpace; :: thesis: for f being PartFunc of S,T st f is_uniformly_continuous_on X holds
p (#) f is_uniformly_continuous_on X
let f be PartFunc of S,T; :: thesis: ( f is_uniformly_continuous_on X implies p (#) f is_uniformly_continuous_on X )
assume A1: f is_uniformly_continuous_on X ; :: thesis: p (#) f is_uniformly_continuous_on X
then X c= dom f by Def1;
hence A2: X c= dom (p (#) f) by VFUNCT_1:def 4; :: according to NFCONT_2:def 1 :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) )

now

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

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

let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) ) )
assume A4: 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) )
then consider s being Real such that
A5: 0 < s and
for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r by A1, Def1;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) )
thus 0 < s by A5; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r
let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r )
assume that
A6: x1 in X and
A7: x2 in X and
||.(x1 - x2).|| < s ; :: thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by A2, A6, VFUNCT_1:def 4
.= ||.((0. T) - ((p (#) f) /. x2)).|| by A3, RLVECT_1:10
.= ||.((0. T) - (p * (f /. x2))).|| by A2, A7, VFUNCT_1:def 4
.= ||.((0. T) - (0. T)).|| by A3, RLVECT_1:10
.= ||.(0. T).|| by RLVECT_1:13
.= 0 by NORMSP_0:def 6 ;
hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r by A4; :: thesis: verum

end;

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

then A9: 0 <> abs p by COMPLEX1:47;
let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) ) )
A10: 0 < abs p by A8, COMPLEX1:47;
assume 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) )
then 0 < r / (abs p) by A10, XREAL_1:139;
then consider s being Real such that
A11: 0 < s and
A12: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r / (abs p) by A1, Def1;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) )
thus 0 < s by A11; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r
let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r )
assume that
A13: x1 in X and
A14: x2 in X and
A15: ||.(x1 - x2).|| < s ; :: thesis: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r
A16: ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| = ||.((p * (f /. x1)) - ((p (#) f) /. x2)).|| by A2, A13, VFUNCT_1:def 4
.= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by A2, A14, VFUNCT_1:def 4
.= ||.(p * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= (abs p) * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;
(abs p) * ||.((f /. x1) - (f /. x2)).|| < (r / (abs p)) * (abs p) by A10, A12, A13, A14, A15, XREAL_1:68;
hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r by A9, A16, 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 S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r ) ) ; :: thesis: verum