A2:
the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) = (cl_Ball o,r) \ {p}
by PRE_TOPC:29;
defpred S1[ set , set ] means ex z being Point of (TOP-REAL n) st
( $1 = z & $2 = HC p,z,o,r );
A3:
for x being set st x in the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) holds
ex y being set st
( y in the carrier of (Tcircle o,r) & S1[x,y] )
proof
let x be
set ;
:: thesis: ( x in the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) implies ex y being set st
( y in the carrier of (Tcircle o,r) & S1[x,y] ) )
assume A4:
x in the
carrier of
((TOP-REAL n) | ((cl_Ball o,r) \ {p}))
;
:: thesis: ex y being set st
( y in the carrier of (Tcircle o,r) & S1[x,y] )
reconsider z =
x as
Point of
(TOP-REAL n) by A4, PRE_TOPC:55;
z in cl_Ball o,
r
by A2, A4, XBOOLE_0:def 5;
then A5:
z is
Point of
(Tdisk o,r)
by BROUWER:3;
p <> z
by A2, A4, ZFMISC_1:64;
then
HC p,
z,
o,
r is
Point of
(Tcircle o,r)
by A1, A5, BROUWER:6;
hence
ex
y being
set st
(
y in the
carrier of
(Tcircle o,r) &
S1[
x,
y] )
;
:: thesis: verum
end;
consider f being Function of the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),the carrier of (Tcircle o,r) such that
A6:
for x being set st x in the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) holds
S1[x,f . x]
from FUNCT_2:sch 1(A3);
reconsider f = f as Function of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),(Tcircle o,r) ;
take
f
; :: thesis: for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC p,y,o,r )
let x be Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})); :: thesis: ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC p,y,o,r )
thus
ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC p,y,o,r )
by A6; :: thesis: verum