let f, g be Function of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})),(Tcircle o,r); :: 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 ) ) & ( for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC p,y,o,r ) ) implies f = g )
assume that
A7: 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 ) and
A8: for x being Point of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC p,y,o,r ) ; :: thesis: f = g

now

let x be set ; :: thesis: ( x in the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) implies f . x = g . x )
assume A9: x in the carrier of ((TOP-REAL n) | ((cl_Ball o,r) \ {p})) ; :: thesis: f . x = g . x
A10: ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC p,y,o,r ) by A7, A9;
ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC p,y,o,r ) by A8, A9;
hence f . x = g . x by A10; :: thesis: verum

end;

hence f = g by FUNCT_2:18; :: thesis: verum