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 ;

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 ) )

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

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 ;

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 ) )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 ) )

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

end;

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 ) )

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 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;( 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 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

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 ) )

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 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 / |.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 / |.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 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

.= |.p.| * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;

|.p.| * ||.((f /. x1) - (f /. x2)).|| < (r / |.p.|) * |.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;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 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 / |.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 / |.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 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

.= |.p.| * ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:def 1 ;

|.p.| * ||.((f /. x1) - (f /. x2)).|| < (r / |.p.|) * |.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

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