let i be non zero Integer; :: thesis: cosec is (2 * PI) * i -periodic
i in INT by INT_1:def 2;
then consider k being Nat such that
A1: ( i = k or i = - k ) by INT_1:def 1;

per cases ( i = k or i = - k ) by A1;

suppose i = k ; :: thesis: cosec is (2 * PI) * i -periodic

then ex m being Nat st i = m + 1 by NAT_1:6;
hence cosec is (2 * PI) * i -periodic by Lm9; :: thesis: verum

end;

suppose A2: i = - k ; :: thesis: cosec is (2 * PI) * i -periodic

then consider m being Nat such that
A3: - i = m + 1 by NAT_1:6;
cosec is (2 * PI) * (m + 1) -periodic by Lm9;
then cosec is - ((2 * PI) * (m + 1)) -periodic by Th14;
hence cosec is (2 * PI) * i -periodic by A2, A3; :: thesis: verum

end;

end;