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:
then X c= dom f ;
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 :: 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 ) )
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;
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
.= ||.((0. T) - ((p (#) f) /. x2)).|| by
.= ||.((0. T) - (p * (f /. x2))).|| by
.= ||.((0. T) - (0. T)).|| by
.= ||.(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 <> |.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 < |.p.| by ;
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 / |.p.| by ;
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 / |.p.| by A1;
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
.= ||.((p * (f /. x1)) - (p * (f /. x2))).|| by
.= ||.(p * ((f /. x1) - (f /. x2))).|| by RLVECT_1:34
.= |.p.| * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;
|.p.| * ||.((f /. x1) - (f /. x2)).|| < (r / |.p.|) * |.p.| by ;
hence ||.(((p (#) f) /. x1) - ((p (#) f) /. x2)).|| < r by ; :: 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