let o, p be Point of (TOP-REAL 2); :: thesis: for r being positive real number st p is Point of (Tdisk o,r) holds
DiskProj o,r,p is continuous
let r be positive real number ; :: thesis: ( p is Point of (Tdisk o,r) implies DiskProj o,r,p is continuous )
assume A1:
p is Point of (Tdisk o,r)
; :: thesis: DiskProj o,r,p is continuous
set D = Tdisk o,r;
set cB = cl_Ball o,r;
set Bp = (cl_Ball o,r) \ {p};
set OK = [:((cl_Ball o,r) \ {p}),{p}:];
set D1 = (TOP-REAL 2) | ((cl_Ball o,r) \ {p});
set D2 = (TOP-REAL 2) | {p};
set S1 = Tcircle o,r;
A2:
p in {p}
by TARSKI:def 1;
A3:
the carrier of (Tdisk o,r) = cl_Ball o,r
by BROUWER:3;
A4:
the carrier of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) = (cl_Ball o,r) \ {p}
by PRE_TOPC:29;
A5:
the carrier of ((TOP-REAL 2) | {p}) = {p}
by PRE_TOPC:29;
set TD = [:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:];
set gg = DiskProj o,r,p;
set xo = diffX2_1 o;
set yo = diffX2_2 o;
set dx = diffX1_X2_1 ;
set dy = diffX1_X2_2 ;
set fx2 = Proj2_1 ;
set fy2 = Proj2_2 ;
reconsider rr = r ^2 as Real by XREAL_0:def 1;
set f1 = the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] --> rr;
reconsider f1 = the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] --> rr as continuous RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] by Lm8;
set Zf1 = f1 | [:((cl_Ball o,r) \ {p}),{p}:];
set Zfx2 = Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:];
set Zfy2 = Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:];
set Zdx = diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:];
set Zdy = diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:];
set Zxo = (diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:];
set Zyo = (diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:];
set xx = ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]);
set yy = ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]);
set m = ((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]));
A6:
the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]) = [:((cl_Ball o,r) \ {p}),{p}:]
by PRE_TOPC:29;
A7:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
(diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
A8:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
(diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
A9:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: (Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
(Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
A10:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: (Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
(Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
A11:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = f1 . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: (f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
(f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
A12:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
A13:
for y being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
proof
let y be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]let z be
Point of
((TOP-REAL 2) | {p});
:: thesis: ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
[y,z] in [:((cl_Ball o,r) \ {p}),{p}:]
by A4, A5, ZFMISC_1:def 2;
hence
((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
by FUNCT_1:72;
:: thesis: verum
end;
now let b be
real number ;
:: thesis: ( b in rng (((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) implies 0 < b )assume
b in rng (((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))
;
:: thesis: 0 < bthen consider a being
set such that A14:
a in dom (((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))
and A15:
(((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) . a = b
by FUNCT_1:def 5;
consider y,
z being
set such that A16:
(
y in (cl_Ball o,r) \ {p} &
z in {p} &
a = [y,z] )
by A6, A14, ZFMISC_1:def 2;
A17:
z = p
by A16, TARSKI:def 1;
reconsider y =
y,
z =
z as
Point of
(TOP-REAL 2) by A16;
A18:
y <> z
by A16, A17, ZFMISC_1:64;
[y,z] = [([y,z] `1 ),([y,z] `2 )]
by Lm7, MCART_1:23;
then A19:
(
y = [y,z] `1 &
z = [y,z] `2 )
by ZFMISC_1:33;
A20:
diffX1_X2_1 . [y,z] = (([y,z] `1 ) `1 ) - (([y,z] `2 ) `1 )
by Def3;
A21:
diffX1_X2_2 . [y,z] = (([y,z] `1 ) `2 ) - (([y,z] `2 ) `2 )
by Def4;
set r1 =
(y `1 ) - (z `1 );
set r2 =
(y `2 ) - (z `2 );
A22:
(diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
by A4, A5, A7, A16;
A23:
(diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
by A4, A5, A8, A16;
A24:
(((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) . [y,z] =
(((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z]) + (((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z])
by A14, A16, VALUED_1:1
.=
(((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z]) * ((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z])) + (((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z])
by VALUED_1:5
.=
(((y `1 ) - (z `1 )) ^2 ) + (((y `2 ) - (z `2 )) ^2 )
by A19, A20, A21, A22, A23, VALUED_1:5
;
hence
0 < b
by A15, A16, A24;
:: thesis: verum end;
then reconsider m = ((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) as continuous positive-yielding RealMap of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]) by PARTFUN3:def 1;
set p1 = ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])));
set p2 = ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:]);
A26:
dom (((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:])) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
now let b be
real number ;
:: thesis: ( b in rng (((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:])) implies 0 >= b )assume
b in rng (((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:]))
;
:: thesis: 0 >= bthen consider a being
set such that A27:
a in dom (((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:]))
and A28:
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:])) . a = b
by FUNCT_1:def 5;
consider y,
z being
set such that A29:
(
y in (cl_Ball o,r) \ {p} &
z in {p} &
a = [y,z] )
by A6, A27, ZFMISC_1:def 2;
reconsider y =
y,
z =
z as
Point of
(TOP-REAL 2) by A29;
set r3 =
(z `1 ) - (o `1 );
set r4 =
(z `2 ) - (o `2 );
A30:
(f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
by A4, A5, A11, A29;
A31:
((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
by A4, A5, A12, A29;
A32:
((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
by A4, A5, A13, A29;
A33:
[y,z] = [([y,z] `1 ),([y,z] `2 )]
by Lm7, MCART_1:23;
A34:
(diffX2_1 o) . [y,z] = (([y,z] `2 ) `1 ) - (o `1 )
by Def1;
A35:
(
y = [y,z] `1 &
z = [y,z] `2 )
by A33, ZFMISC_1:33;
A36:
(diffX2_2 o) . [y,z] = (([y,z] `2 ) `2 ) - (o `2 )
by Def2;
A37:
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z] =
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) . [y,z]) - ((f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z])
by A27, A29, VALUED_1:13
.=
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) . [y,z]) - (r ^2 )
by A30, FUNCOP_1:13
.=
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z]) + ((((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z])) - (r ^2 )
by A27, A29, VALUED_1:1
.=
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z]) * (((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,z])) + ((((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,z])) - (r ^2 )
by VALUED_1:5
.=
((((z `1 ) - (o `1 )) ^2 ) + (((z `2 ) - (o `2 )) ^2 )) - (r ^2 )
by A31, A32, A34, A35, A36, VALUED_1:5
;
z = p
by A29, TARSKI:def 1;
then
|.(z - o).| <= r
by A1, A3, TOPREAL9:8;
then A38:
|.(z - o).| ^2 <= r ^2
by SQUARE_1:77;
|.(z - o).| ^2 =
(((z - o) `1 ) ^2 ) + (((z - o) `2 ) ^2 )
by JGRAPH_1:46
.=
(((z `1 ) - (o `1 )) ^2 ) + (((z - o) `2 ) ^2 )
by TOPREAL3:8
.=
(((z `1 ) - (o `1 )) ^2 ) + (((z `2 ) - (o `2 )) ^2 )
by TOPREAL3:8
;
then
((((z `1 ) - (o `1 )) ^2 ) + (((z `2 ) - (o `2 )) ^2 )) - (r ^2 ) <= (r ^2 ) - (r ^2 )
by A38, XREAL_1:11;
hence
0 >= b
by A28, A29, A37;
:: thesis: verum end;
then reconsider p2 = ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) - (f1 | [:((cl_Ball o,r) \ {p}),{p}:]) as continuous nonpositive-yielding RealMap of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]) by PARTFUN3:def 3;
set pp = (((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2);
set k = ((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m;
set x3 = (Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]));
set y3 = (Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]));
reconsider X3 = (Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])), Y3 = (Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) as Function of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]),R^1 by TOPMETR:24;
set F = <:X3,Y3:>;
set R = R2Homeomorphism ;
A39:
for x being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) holds (DiskProj o,r,p) . x = (R2Homeomorphism * <:X3,Y3:>) . [x,p]
proof
let x be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: (DiskProj o,r,p) . x = (R2Homeomorphism * <:X3,Y3:>) . [x,p]
consider y being
Point of
(TOP-REAL 2) such that A40:
(
x = y &
(DiskProj o,r,p) . x = HC p,
y,
o,
r )
by A1, Def7;
A41:
x <> p
by A4, ZFMISC_1:64;
A42:
[y,p] in [:((cl_Ball o,r) \ {p}),{p}:]
by A2, A4, A40, ZFMISC_1:def 2;
set r1 =
(y `1 ) - (p `1 );
set r2 =
(y `2 ) - (p `2 );
set r3 =
(p `1 ) - (o `1 );
set r4 =
(p `2 ) - (o `2 );
set l =
((- ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 ))))) + (sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )))))) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ));
[y,p] = [([y,p] `1 ),([y,p] `2 )]
by Lm7, MCART_1:23;
then A43:
(
y = [y,p] `1 &
p = [y,p] `2 )
by ZFMISC_1:33;
A44:
Proj2_1 . [y,p] = ([y,p] `2 ) `1
by Def5;
A45:
Proj2_2 . [y,p] = ([y,p] `2 ) `2
by Def6;
A46:
diffX1_X2_1 . [y,p] = (([y,p] `1 ) `1 ) - (([y,p] `2 ) `1 )
by Def3;
A47:
diffX1_X2_2 . [y,p] = (([y,p] `1 ) `2 ) - (([y,p] `2 ) `2 )
by Def4;
A48:
(diffX2_1 o) . [y,p] = (([y,p] `2 ) `1 ) - (o `1 )
by Def1;
A49:
(diffX2_2 o) . [y,p] = (([y,p] `2 ) `2 ) - (o `2 )
by Def2;
A50:
dom X3 = the
carrier of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
A51:
dom Y3 = the
carrier of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
A52:
dom ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)) = the
carrier of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
A53:
(
y is
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) &
p is
Point of
((TOP-REAL 2) | {p}) )
by A5, A40, TARSKI:def 1;
then A54:
(diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p] = diffX1_X2_1 . [y,p]
by A7;
A55:
(diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p] = diffX1_X2_2 . [y,p]
by A8, A53;
A56:
(f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p] = f1 . [y,p]
by A11, A53;
A57:
((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p] = (diffX2_1 o) . [y,p]
by A12, A53;
A58:
((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p] = (diffX2_2 o) . [y,p]
by A13, A53;
A59:
m . [y,p] =
(((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p]) + (((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by A6, A42, VALUED_1:1
.=
(((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p]) * ((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p])) + (((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by VALUED_1:5
.=
(((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )
by A43, A46, A47, A54, A55, VALUED_1:5
;
A60:
(((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p] = (((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p]) * ((diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p])
by VALUED_1:5;
A61:
(((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p] = (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p]) * ((diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p])
by VALUED_1:5;
A62:
((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) . [y,p] = ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p]) + ((((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by A6, A42, VALUED_1:1;
then A63:
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) . [y,p] = ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2
by A43, A46, A47, A48, A49, A54, A55, A57, A58, A60, A61, VALUED_1:5;
A64:
p2 . [y,p] =
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) . [y,p]) - ((f1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p])
by A6, A26, A42, VALUED_1:13
.=
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]))) . [y,p]) - (r ^2 )
by A56, FUNCOP_1:13
.=
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p]) + ((((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])) - (r ^2 )
by A6, A42, VALUED_1:1
.=
(((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p]) * (((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p])) + ((((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])) - (r ^2 )
by VALUED_1:5
.=
((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )
by A43, A48, A49, A57, A58, VALUED_1:5
;
dom (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2))) = the
carrier of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
then A65:
(sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2))) . [y,p] =
sqrt (((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)) . [y,p])
by A6, A42, PARTFUN3:def 5
.=
sqrt (((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) . [y,p]) - ((m (#) p2) . [y,p]))
by A6, A42, A52, VALUED_1:13
.=
sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 ))))
by A59, A63, A64, VALUED_1:5
;
dom (((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) = the
carrier of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
then A66:
(((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) . [y,p] =
(((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) . [y,p]) * ((m . [y,p]) " )
by A6, A42, RFUNCT_1:def 4
.=
(((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) . [y,p]) / (m . [y,p])
by XCMPLX_0:def 9
.=
(((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) . [y,p]) + ((sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2))) . [y,p])) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ))
by A6, A42, A59, VALUED_1:1
.=
((- ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 ))))) + (sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )))))) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ))
by A43, A46, A47, A48, A49, A54, A55, A57, A58, A60, A61, A62, A65, VALUED_1:8
;
A67:
X3 . [y,p] =
((Proj2_1 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p]) + (((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by A6, A42, VALUED_1:1
.=
(p `1 ) + (((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by A9, A43, A44, A53
.=
(p `1 ) + ((((- ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 ))))) + (sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )))))) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ))) * ((y `1 ) - (p `1 )))
by A43, A46, A54, A66, VALUED_1:5
;
A68:
Y3 . [y,p] =
((Proj2_2 | [:((cl_Ball o,r) \ {p}),{p}:]) . [y,p]) + (((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by A6, A42, VALUED_1:1
.=
(p `2 ) + (((((- ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball o,r) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball o,r) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball o,r) \ {p}),{p}:])) . [y,p])
by A10, A43, A45, A53
.=
(p `2 ) + ((((- ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 ))))) + (sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )))))) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ))) * ((y `2 ) - (p `2 )))
by A43, A47, A55, A66, VALUED_1:5
;
A69:
y in (cl_Ball o,r) \ {p}
by A4, A40;
(cl_Ball o,r) \ {p} c= cl_Ball o,
r
by XBOOLE_1:36;
hence (DiskProj o,r,p) . x =
|[((p `1 ) + ((((- ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 ))))) + (sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )))))) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ))) * ((y `1 ) - (p `1 )))),((p `2 ) + ((((- ((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 ))))) + (sqrt ((((((p `1 ) - (o `1 )) * ((y `1 ) - (p `1 ))) + (((p `2 ) - (o `2 )) * ((y `2 ) - (p `2 )))) ^2 ) - (((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 )) * (((((p `1 ) - (o `1 )) ^2 ) + (((p `2 ) - (o `2 )) ^2 )) - (r ^2 )))))) / ((((y `1 ) - (p `1 )) ^2 ) + (((y `2 ) - (p `2 )) ^2 ))) * ((y `2 ) - (p `2 ))))]|
by A1, A3, A40, A41, A69, BROUWER:8
.=
R2Homeomorphism . [(X3 . [y,p]),(Y3 . [y,p])]
by A67, A68, TOPREALA:def 2
.=
R2Homeomorphism . (<:X3,Y3:> . [y,p])
by A6, A42, A50, A51, FUNCT_3:69
.=
(R2Homeomorphism * <:X3,Y3:>) . [x,p]
by A6, A40, A42, FUNCT_2:21
;
:: thesis: verum
end;
( X3 is continuous & Y3 is continuous )
by TOPREAL6:83;
then reconsider F = <:X3,Y3:> as continuous Function of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]),[:R^1 ,R^1 :] by YELLOW12:41;
for pp being Point of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))
for V being Subset of (Tcircle o,r) st (DiskProj o,r,p) . pp in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) st
( pp in W & W is open & (DiskProj o,r,p) .: W c= V )
proof
let pp be
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p}));
:: thesis: for V being Subset of (Tcircle o,r) st (DiskProj o,r,p) . pp in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) st
( pp in W & W is open & (DiskProj o,r,p) .: W c= V )let V be
Subset of
(Tcircle o,r);
:: thesis: ( (DiskProj o,r,p) . pp in V & V is open implies ex W being Subset of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) st
( pp in W & W is open & (DiskProj o,r,p) .: W c= V ) )
assume that A70:
(DiskProj o,r,p) . pp in V
and A71:
V is
open
;
:: thesis: ex W being Subset of ((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) st
( pp in W & W is open & (DiskProj o,r,p) .: W c= V )
reconsider p1 =
pp,
fp =
p as
Point of
(TOP-REAL 2) by PRE_TOPC:55;
A72:
[pp,p] in [:((cl_Ball o,r) \ {p}),{p}:]
by A2, A4, ZFMISC_1:def 2;
consider V1 being
Subset of
(TOP-REAL 2) such that A73:
V1 is
open
and A74:
V1 /\ ([#] (Tcircle o,r)) = V
by A71, TOPS_2:32;
A75:
(DiskProj o,r,p) . pp = (R2Homeomorphism * F) . [pp,p]
by A39;
R2Homeomorphism " is
being_homeomorphism
by TOPREALA:56, TOPS_2:70;
then A76:
(R2Homeomorphism " ) .: V1 is
open
by A73, TOPGRP_1:25;
A77:
dom F = the
carrier of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])
by FUNCT_2:def 1;
A78:
dom R2Homeomorphism = the
carrier of
[:R^1 ,R^1 :]
by FUNCT_2:def 1;
then A79:
rng F c= dom R2Homeomorphism
;
then A80:
dom (R2Homeomorphism * F) = dom F
by RELAT_1:46;
A81:
(
rng R2Homeomorphism = [#] (TOP-REAL 2) &
R2Homeomorphism is
one-to-one )
by TOPREALA:56, TOPS_2:def 5;
A82:
(R2Homeomorphism " ) * (R2Homeomorphism * F) =
((R2Homeomorphism " ) * R2Homeomorphism ) * F
by RELAT_1:55
.=
(id (dom R2Homeomorphism )) * F
by A81, TOPS_2:65
;
dom (id (dom R2Homeomorphism )) = dom R2Homeomorphism
by RELAT_1:71;
then A83:
dom ((id (dom R2Homeomorphism )) * F) = dom F
by A79, RELAT_1:46;
for
x being
set st
x in dom F holds
((id (dom R2Homeomorphism )) * F) . x = F . x
then A86:
(id (dom R2Homeomorphism )) * F = F
by A83, FUNCT_1:9;
(R2Homeomorphism * F) . [p1,fp] in V1
by A70, A74, A75, XBOOLE_0:def 4;
then
(R2Homeomorphism " ) . ((R2Homeomorphism * F) . [p1,fp]) in (R2Homeomorphism " ) .: V1
by FUNCT_2:43;
then
((R2Homeomorphism " ) * (R2Homeomorphism * F)) . [p1,fp] in (R2Homeomorphism " ) .: V1
by A6, A72, A77, A80, FUNCT_1:23;
then consider W being
Subset of
([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]) such that A87:
[p1,fp] in W
and A88:
W is
open
and A89:
F .: W c= (R2Homeomorphism " ) .: V1
by A6, A72, A76, A82, A86, JGRAPH_2:20;
consider WW being
Subset of
[:(TOP-REAL 2),(TOP-REAL 2):] such that A90:
WW is
open
and A91:
WW /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])) = W
by A88, TOPS_2:32;
consider SF being
Subset-Family of
[:(TOP-REAL 2),(TOP-REAL 2):] such that A92:
WW = union SF
and A93:
for
e being
set st
e in SF holds
ex
X1,
Y1 being
Subset of
(TOP-REAL 2) st
(
e = [:X1,Y1:] &
X1 is
open &
Y1 is
open )
by A90, BORSUK_1:45;
[p1,fp] in WW
by A87, A91, XBOOLE_0:def 4;
then consider Z being
set such that A94:
[p1,fp] in Z
and A95:
Z in SF
by A92, TARSKI:def 4;
consider X1,
Y1 being
Subset of
(TOP-REAL 2) such that A96:
Z = [:X1,Y1:]
and A97:
X1 is
open
and
Y1 is
open
by A93, A95;
set ZZ =
Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]));
reconsider XX =
X1 /\ ([#] ((TOP-REAL 2) | ((cl_Ball o,r) \ {p}))) as
open Subset of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) by A97, TOPS_2:32;
take
XX
;
:: thesis: ( pp in XX & XX is open & (DiskProj o,r,p) .: XX c= V )
pp in X1
by A94, A96, ZFMISC_1:106;
hence
pp in XX
by XBOOLE_0:def 4;
:: thesis: ( XX is open & (DiskProj o,r,p) .: XX c= V )
thus
XX is
open
;
:: thesis: (DiskProj o,r,p) .: XX c= V
let b be
set ;
:: according to TARSKI:def 3 :: thesis: ( not b in (DiskProj o,r,p) .: XX or b in V )
assume
b in (DiskProj o,r,p) .: XX
;
:: thesis: b in V
then consider a being
Point of
((TOP-REAL 2) | ((cl_Ball o,r) \ {p})) such that A98:
a in XX
and A99:
b = (DiskProj o,r,p) . a
by FUNCT_2:116;
reconsider a1 =
a,
fa =
fp as
Point of
(TOP-REAL 2) by PRE_TOPC:55;
A100:
a in X1
by A98, XBOOLE_0:def 4;
A101:
[a,p] in [:((cl_Ball o,r) \ {p}),{p}:]
by A2, A4, ZFMISC_1:def 2;
fa in Y1
by A94, A96, ZFMISC_1:106;
then
[a,fa] in Z
by A96, A100, ZFMISC_1:def 2;
then
[a,fa] in Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]))
by A6, A101, XBOOLE_0:def 4;
then A102:
F . [a1,fa] in F .: (Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])))
by FUNCT_2:43;
A103:
R2Homeomorphism " = R2Homeomorphism "
by A81, TOPS_2:def 4;
A104:
dom (R2Homeomorphism " ) = [#] (TOP-REAL 2)
by A81, TOPS_2:62;
Z c= WW
by A92, A95, ZFMISC_1:92;
then
Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:])) c= WW /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]))
by XBOOLE_1:27;
then
F .: (Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball o,r) \ {p}),{p}:]))) c= F .: W
by A91, RELAT_1:156;
then
F . [a1,fa] in F .: W
by A102;
then
R2Homeomorphism . (F . [a1,fa]) in R2Homeomorphism .: ((R2Homeomorphism " ) .: V1)
by A89, FUNCT_2:43;
then
(R2Homeomorphism * F) . [a1,fa] in R2Homeomorphism .: ((R2Homeomorphism " ) .: V1)
by A6, A101, FUNCT_2:21;
then
(R2Homeomorphism * F) . [a1,fa] in V1
by A81, A103, A104, PARTFUN3:1;
then
(DiskProj o,r,p) . a in V1
by A39;
hence
b in V
by A74, A99, XBOOLE_0:def 4;
:: thesis: verum
end;
hence
DiskProj o,r,p is continuous
by JGRAPH_2:20; :: thesis: verum