let n be Nat; :: thesis: for p being Point of (TOP-REAL n)

for r being positive Real

for f being Function of (Tunit_ball n),(Tball (p,r)) st n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) holds

f is being_homeomorphism

let p be Point of (TOP-REAL n); :: thesis: for r being positive Real

for f being Function of (Tunit_ball n),(Tball (p,r)) st n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) holds

f is being_homeomorphism

let r be positive Real; :: thesis: for f being Function of (Tunit_ball n),(Tball (p,r)) st n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) holds

f is being_homeomorphism

let f be Function of (Tunit_ball n),(Tball (p,r)); :: thesis: ( n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) implies f is being_homeomorphism )

assume that

A1: n <> 0 and

A2: for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ; :: thesis: f is being_homeomorphism

reconsider n1 = n as non zero Element of NAT by A1, ORDINAL1:def 12;

reconsider x = p as Point of (TOP-REAL n1) ;

defpred S_{1}[ Point of (TOP-REAL n1), set ] means $2 = (r * $1) + x;

set U = Tunit_ball n;

set B = Tball (x,r);

A3: for u being Point of (TOP-REAL n1) ex y being Point of (TOP-REAL n1) st S_{1}[u,y]
;

consider F being Function of (TOP-REAL n1),(TOP-REAL n1) such that

A4: for x being Point of (TOP-REAL n1) holds S_{1}[x,F . x]
from FUNCT_2:sch 3(A3);

defpred S_{2}[ Point of (TOP-REAL n1), set ] means $2 = (1 / r) * ($1 - x);

A5: for u being Point of (TOP-REAL n1) ex y being Point of (TOP-REAL n1) st S_{2}[u,y]
;

consider G being Function of (TOP-REAL n1),(TOP-REAL n1) such that

A6: for a being Point of (TOP-REAL n1) holds S_{2}[a,G . a]
from FUNCT_2:sch 3(A5);

set f2 = (TOP-REAL n1) --> x;

set f1 = id (TOP-REAL n1);

dom G = the carrier of (TOP-REAL n) by FUNCT_2:def 1;

then A7: dom (G | (Ball (x,r))) = Ball (x,r) by RELAT_1:62;

for p being Point of (TOP-REAL n1) holds G . p = ((1 / r) * ((id (TOP-REAL n1)) . p)) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p))

A9: dom f = [#] (Tunit_ball n) by FUNCT_2:def 1;

A10: dom f = the carrier of (Tunit_ball n) by FUNCT_2:def 1;

for p being Point of (TOP-REAL n1) holds F . p = (r * ((id (TOP-REAL n1)) . p)) + (1 * (((TOP-REAL n1) --> x) . p))

A12: the carrier of (Tball (x,r)) = Ball (x,r) by MFOLD_0:2;

A13: the carrier of (Tunit_ball n) = Ball ((0. (TOP-REAL n)),1) by MFOLD_0:2;

A14: for a being object st a in dom f holds

f . a = (F | (Ball ((0. (TOP-REAL n)),1))) . a

A17: rng f = [#] (Tball (x,r))

A26: for a being object st a in dom (f ") holds

(f ") . a = (G | (Ball (x,r))) . a

then dom (F | (Ball ((0. (TOP-REAL n)),1))) = Ball ((0. (TOP-REAL n)),1) by RELAT_1:62;

then A31: f is continuous by A13, A10, A11, A14, BORSUK_4:44, FUNCT_1:2;

A32: dom (f ") = the carrier of (Tball (x,r)) by FUNCT_2:def 1;

f " is continuous by A32, A12, A7, A8, A26, BORSUK_4:44, FUNCT_1:2;

hence f is being_homeomorphism by A9, A17, A25, A31, TOPS_2:def 5; :: thesis: verum

for r being positive Real

for f being Function of (Tunit_ball n),(Tball (p,r)) st n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) holds

f is being_homeomorphism

let p be Point of (TOP-REAL n); :: thesis: for r being positive Real

for f being Function of (Tunit_ball n),(Tball (p,r)) st n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) holds

f is being_homeomorphism

let r be positive Real; :: thesis: for f being Function of (Tunit_ball n),(Tball (p,r)) st n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) holds

f is being_homeomorphism

let f be Function of (Tunit_ball n),(Tball (p,r)); :: thesis: ( n <> 0 & ( for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ) implies f is being_homeomorphism )

assume that

A1: n <> 0 and

A2: for a being Point of (Tunit_ball n)

for b being Point of (TOP-REAL n) st a = b holds

f . a = (r * b) + p ; :: thesis: f is being_homeomorphism

reconsider n1 = n as non zero Element of NAT by A1, ORDINAL1:def 12;

reconsider x = p as Point of (TOP-REAL n1) ;

defpred S

set U = Tunit_ball n;

set B = Tball (x,r);

A3: for u being Point of (TOP-REAL n1) ex y being Point of (TOP-REAL n1) st S

consider F being Function of (TOP-REAL n1),(TOP-REAL n1) such that

A4: for x being Point of (TOP-REAL n1) holds S

defpred S

A5: for u being Point of (TOP-REAL n1) ex y being Point of (TOP-REAL n1) st S

consider G being Function of (TOP-REAL n1),(TOP-REAL n1) such that

A6: for a being Point of (TOP-REAL n1) holds S

set f2 = (TOP-REAL n1) --> x;

set f1 = id (TOP-REAL n1);

dom G = the carrier of (TOP-REAL n) by FUNCT_2:def 1;

then A7: dom (G | (Ball (x,r))) = Ball (x,r) by RELAT_1:62;

for p being Point of (TOP-REAL n1) holds G . p = ((1 / r) * ((id (TOP-REAL n1)) . p)) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p))

proof

then A8:
G is continuous
by TOPALG_1:16;
let p be Point of (TOP-REAL n1); :: thesis: G . p = ((1 / r) * ((id (TOP-REAL n1)) . p)) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p))

thus ((1 / r) * ((id (TOP-REAL n1)) . p)) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p)) = ((1 / r) * p) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p))

.= ((1 / r) * p) + ((- (1 / r)) * x) by FUNCOP_1:7

.= ((1 / r) * p) - ((1 / r) * x) by RLVECT_1:79

.= (1 / r) * (p - x) by RLVECT_1:34

.= G . p by A6 ; :: thesis: verum

end;thus ((1 / r) * ((id (TOP-REAL n1)) . p)) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p)) = ((1 / r) * p) + ((- (1 / r)) * (((TOP-REAL n1) --> x) . p))

.= ((1 / r) * p) + ((- (1 / r)) * x) by FUNCOP_1:7

.= ((1 / r) * p) - ((1 / r) * x) by RLVECT_1:79

.= (1 / r) * (p - x) by RLVECT_1:34

.= G . p by A6 ; :: thesis: verum

A9: dom f = [#] (Tunit_ball n) by FUNCT_2:def 1;

A10: dom f = the carrier of (Tunit_ball n) by FUNCT_2:def 1;

for p being Point of (TOP-REAL n1) holds F . p = (r * ((id (TOP-REAL n1)) . p)) + (1 * (((TOP-REAL n1) --> x) . p))

proof

then A11:
F is continuous
by TOPALG_1:16;
let p be Point of (TOP-REAL n1); :: thesis: F . p = (r * ((id (TOP-REAL n1)) . p)) + (1 * (((TOP-REAL n1) --> x) . p))

thus (r * ((id (TOP-REAL n1)) . p)) + (1 * (((TOP-REAL n1) --> x) . p)) = (r * ((id (TOP-REAL n1)) . p)) + (((TOP-REAL n1) --> x) . p) by RLVECT_1:def 8

.= (r * p) + (((TOP-REAL n1) --> x) . p)

.= (r * p) + x by FUNCOP_1:7

.= F . p by A4 ; :: thesis: verum

end;thus (r * ((id (TOP-REAL n1)) . p)) + (1 * (((TOP-REAL n1) --> x) . p)) = (r * ((id (TOP-REAL n1)) . p)) + (((TOP-REAL n1) --> x) . p) by RLVECT_1:def 8

.= (r * p) + (((TOP-REAL n1) --> x) . p)

.= (r * p) + x by FUNCOP_1:7

.= F . p by A4 ; :: thesis: verum

A12: the carrier of (Tball (x,r)) = Ball (x,r) by MFOLD_0:2;

A13: the carrier of (Tunit_ball n) = Ball ((0. (TOP-REAL n)),1) by MFOLD_0:2;

A14: for a being object st a in dom f holds

f . a = (F | (Ball ((0. (TOP-REAL n)),1))) . a

proof

A16:
(1 / r) * r = 1
by XCMPLX_1:87;
let a be object ; :: thesis: ( a in dom f implies f . a = (F | (Ball ((0. (TOP-REAL n)),1))) . a )

assume A15: a in dom f ; :: thesis: f . a = (F | (Ball ((0. (TOP-REAL n)),1))) . a

reconsider y = a as Point of (TOP-REAL n1) by A15, PRE_TOPC:25;

thus f . a = (r * y) + x by A2, A15

.= F . y by A4

.= (F | (Ball ((0. (TOP-REAL n)),1))) . a by A13, A15, FUNCT_1:49 ; :: thesis: verum

end;assume A15: a in dom f ; :: thesis: f . a = (F | (Ball ((0. (TOP-REAL n)),1))) . a

reconsider y = a as Point of (TOP-REAL n1) by A15, PRE_TOPC:25;

thus f . a = (r * y) + x by A2, A15

.= F . y by A4

.= (F | (Ball ((0. (TOP-REAL n)),1))) . a by A13, A15, FUNCT_1:49 ; :: thesis: verum

A17: rng f = [#] (Tball (x,r))

proof

hence rng f = [#] (Tball (x,r)) by XBOOLE_0:def 10; :: thesis: verum

end;

now :: thesis: for b being object st b in [#] (Tball (x,r)) holds

b in rng f

then
[#] (Tball (x,r)) c= rng f
;b in rng f

let b be object ; :: thesis: ( b in [#] (Tball (x,r)) implies b in rng f )

assume A18: b in [#] (Tball (x,r)) ; :: thesis: b in rng f

then reconsider c = b as Point of (TOP-REAL n1) by PRE_TOPC:25;

reconsider r1 = 1 / r as Real ;

set a = r1 * (c - x);

A19: |.((r1 * (c - x)) - (0. (TOP-REAL n1))).| = |.(r1 * (c - x)).| by RLVECT_1:13

.= |.r1.| * |.(c - x).| by TOPRNS_1:7

.= r1 * |.(c - x).| by ABSVALUE:def 1 ;

(1 / r) * |.(c - x).| < (1 / r) * r by XREAL_1:68, A12, A18, TOPREAL9:7;

then A20: r1 * (c - x) in Ball ((0. (TOP-REAL n)),1) by A16, A19;

then f . (r1 * (c - x)) = (r * (r1 * (c - x))) + x by A2, A13

.= ((r * (1 / r)) * (c - x)) + x by RLVECT_1:def 7

.= (c - x) + x by A16, RLVECT_1:def 8

.= b by RLVECT_4:1 ;

hence b in rng f by A13, A10, A20, FUNCT_1:def 3; :: thesis: verum

end;assume A18: b in [#] (Tball (x,r)) ; :: thesis: b in rng f

then reconsider c = b as Point of (TOP-REAL n1) by PRE_TOPC:25;

reconsider r1 = 1 / r as Real ;

set a = r1 * (c - x);

A19: |.((r1 * (c - x)) - (0. (TOP-REAL n1))).| = |.(r1 * (c - x)).| by RLVECT_1:13

.= |.r1.| * |.(c - x).| by TOPRNS_1:7

.= r1 * |.(c - x).| by ABSVALUE:def 1 ;

(1 / r) * |.(c - x).| < (1 / r) * r by XREAL_1:68, A12, A18, TOPREAL9:7;

then A20: r1 * (c - x) in Ball ((0. (TOP-REAL n)),1) by A16, A19;

then f . (r1 * (c - x)) = (r * (r1 * (c - x))) + x by A2, A13

.= ((r * (1 / r)) * (c - x)) + x by RLVECT_1:def 7

.= (c - x) + x by A16, RLVECT_1:def 8

.= b by RLVECT_4:1 ;

hence b in rng f by A13, A10, A20, FUNCT_1:def 3; :: thesis: verum

hence rng f = [#] (Tball (x,r)) by XBOOLE_0:def 10; :: thesis: verum

now :: thesis: for a, b being object st a in dom f & b in dom f & f . a = f . b holds

a = b

then A25:
f is one-to-one
;a = b

let a, b be object ; :: thesis: ( a in dom f & b in dom f & f . a = f . b implies a = b )

assume that

A21: a in dom f and

A22: b in dom f and

A23: f . a = f . b ; :: thesis: a = b

reconsider a1 = a, b1 = b as Point of (TOP-REAL n1) by A13, A10, A21, A22;

A24: f . b1 = (r * b1) + x by A2, A22;

f . a1 = (r * a1) + x by A2, A21;

then r * a1 = ((r * b1) + x) - x by A23, A24, RLVECT_4:1;

hence a = b by RLVECT_1:36, RLVECT_4:1; :: thesis: verum

end;assume that

A21: a in dom f and

A22: b in dom f and

A23: f . a = f . b ; :: thesis: a = b

reconsider a1 = a, b1 = b as Point of (TOP-REAL n1) by A13, A10, A21, A22;

A24: f . b1 = (r * b1) + x by A2, A22;

f . a1 = (r * a1) + x by A2, A21;

then r * a1 = ((r * b1) + x) - x by A23, A24, RLVECT_4:1;

hence a = b by RLVECT_1:36, RLVECT_4:1; :: thesis: verum

A26: for a being object st a in dom (f ") holds

(f ") . a = (G | (Ball (x,r))) . a

proof

dom F = the carrier of (TOP-REAL n)
by FUNCT_2:def 1;
reconsider ff = f as Function ;

let a be object ; :: thesis: ( a in dom (f ") implies (f ") . a = (G | (Ball (x,r))) . a )

assume A27: a in dom (f ") ; :: thesis: (f ") . a = (G | (Ball (x,r))) . a

reconsider y = a as Point of (TOP-REAL n1) by A27, PRE_TOPC:25;

reconsider r1 = 1 / r as Real ;

set e = (1 / r) * (y - x);

A28: f is onto by A17, FUNCT_2:def 3;

A29: |.(((1 / r) * (y - x)) - (0. (TOP-REAL n1))).| = |.((1 / r) * (y - x)).| by RLVECT_1:13

.= |.r1.| * |.(y - x).| by TOPRNS_1:7

.= r1 * |.(y - x).| by ABSVALUE:def 1 ;

(1 / r) * |.(y - x).| < (1 / r) * r by XREAL_1:68, A27, A12, TOPREAL9:7;

then A30: (1 / r) * (y - x) in Ball ((0. (TOP-REAL n)),1) by A16, A29;

then f . ((1 / r) * (y - x)) = (r * ((1 / r) * (y - x))) + x by A2, A13

.= ((r * (1 / r)) * (y - x)) + x by RLVECT_1:def 7

.= (y - x) + x by A16, RLVECT_1:def 8

.= y by RLVECT_4:1 ;

then (ff ") . a = (1 / r) * (y - x) by A13, A10, A25, A30, FUNCT_1:32;

hence (f ") . a = (1 / r) * (y - x) by A28, A25, TOPS_2:def 4

.= G . y by A6

.= (G | (Ball (x,r))) . a by A12, A27, FUNCT_1:49 ;

:: thesis: verum

end;let a be object ; :: thesis: ( a in dom (f ") implies (f ") . a = (G | (Ball (x,r))) . a )

assume A27: a in dom (f ") ; :: thesis: (f ") . a = (G | (Ball (x,r))) . a

reconsider y = a as Point of (TOP-REAL n1) by A27, PRE_TOPC:25;

reconsider r1 = 1 / r as Real ;

set e = (1 / r) * (y - x);

A28: f is onto by A17, FUNCT_2:def 3;

A29: |.(((1 / r) * (y - x)) - (0. (TOP-REAL n1))).| = |.((1 / r) * (y - x)).| by RLVECT_1:13

.= |.r1.| * |.(y - x).| by TOPRNS_1:7

.= r1 * |.(y - x).| by ABSVALUE:def 1 ;

(1 / r) * |.(y - x).| < (1 / r) * r by XREAL_1:68, A27, A12, TOPREAL9:7;

then A30: (1 / r) * (y - x) in Ball ((0. (TOP-REAL n)),1) by A16, A29;

then f . ((1 / r) * (y - x)) = (r * ((1 / r) * (y - x))) + x by A2, A13

.= ((r * (1 / r)) * (y - x)) + x by RLVECT_1:def 7

.= (y - x) + x by A16, RLVECT_1:def 8

.= y by RLVECT_4:1 ;

then (ff ") . a = (1 / r) * (y - x) by A13, A10, A25, A30, FUNCT_1:32;

hence (f ") . a = (1 / r) * (y - x) by A28, A25, TOPS_2:def 4

.= G . y by A6

.= (G | (Ball (x,r))) . a by A12, A27, FUNCT_1:49 ;

:: thesis: verum

then dom (F | (Ball ((0. (TOP-REAL n)),1))) = Ball ((0. (TOP-REAL n)),1) by RELAT_1:62;

then A31: f is continuous by A13, A10, A11, A14, BORSUK_4:44, FUNCT_1:2;

A32: dom (f ") = the carrier of (Tball (x,r)) by FUNCT_2:def 1;

f " is continuous by A32, A12, A7, A8, A26, BORSUK_4:44, FUNCT_1:2;

hence f is being_homeomorphism by A9, A17, A25, A31, TOPS_2:def 5; :: thesis: verum