let i be non zero Integer; :: thesis: |.sin.| + |.cos.| is (PI / 2) * 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: |.sin.| + |.cos.| is (PI / 2) * i -periodic

then ex m being Nat st i = m + 1 by NAT_1:6;
hence |.sin.| + |.cos.| is (PI / 2) * i -periodic by Lm19; :: thesis: verum

end;

suppose A2: i = - k ; :: thesis: |.sin.| + |.cos.| is (PI / 2) * i -periodic

then consider m being Nat such that
A3: - i = m + 1 by NAT_1:6;
|.sin.| + |.cos.| is (PI / 2) * (m + 1) -periodic by Lm19;
then |.sin.| + |.cos.| is - ((PI / 2) * (m + 1)) -periodic by Th14;
hence |.sin.| + |.cos.| is (PI / 2) * i -periodic by A2, A3; :: thesis: verum

end;

end;