let a, b, r be Real; :: thesis: for t being Point of (TOP-REAL 2) holds
( t in inside_of_circle (a,b,r) iff |.(t - |[a,b]|).| < r )
let t be Point of (TOP-REAL 2); :: thesis: ( t in inside_of_circle (a,b,r) iff |.(t - |[a,b]|).| < r )
A1: inside_of_circle (a,b,r) = { x where x is Point of (TOP-REAL 2) : |.(x - |[a,b]|).| < r } by JGRAPH_6:def 6;

hereby :: thesis: ( |.(t - |[a,b]|).| < r implies t in inside_of_circle (a,b,r) )

assume t in inside_of_circle (a,b,r) ; :: thesis: |.(t - |[a,b]|).| < r
then ex x being Point of (TOP-REAL 2) st
( t = x & |.(x - |[a,b]|).| < r ) by A1;
hence |.(t - |[a,b]|).| < r ; :: thesis: verum

end;

thus ( |.(t - |[a,b]|).| < r implies t in inside_of_circle (a,b,r) ) by A1; :: thesis: verum