A1: the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) = (cl_Ball (o,r)) \ {p} by PRE_TOPC:8;
defpred S₁[ set , set ] means ex z being Point of (TOP-REAL n) st
( $1 = z & $2 = HC (p,z,o,r) );
A2: 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 A3: 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 A3, PRE_TOPC:25;
z in cl_Ball (o,r) by A1, A3, XBOOLE_0:def 5;
then A4: z is Point of (Tdisk (o,r)) by BROUWER:3;
p <> z by A1, A3, ZFMISC_1:56;
then HC (p,z,o,r) is Point of (Tcircle (o,r)) by B1, A4, 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
A5: 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(A2);
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 A5; :: thesis: verum