per cases
( r is positive or not r is positive )
;

end;

suppose
r is positive
; :: thesis: Tcircle (x,r) is pathwise_connected

then reconsider r = r as positive Real ;

Tcircle (x,r) is pathwise_connected ;

hence Tcircle (x,r) is pathwise_connected ; :: thesis: verum

end;Tcircle (x,r) is pathwise_connected ;

hence Tcircle (x,r) is pathwise_connected ; :: thesis: verum

suppose
not r is positive
; :: thesis: Tcircle (x,r) is pathwise_connected

then reconsider r = r as non positive non negative Real ;

Tcircle (x,r) is trivial ;

hence Tcircle (x,r) is pathwise_connected ; :: thesis: verum

end;Tcircle (x,r) is trivial ;

hence Tcircle (x,r) is pathwise_connected ; :: thesis: verum