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