let k be Nat; :: thesis: |.sin .| + |.cos .| is (PI / 2) * (k + 1) -periodic
defpred S₁[ Nat] means |.sin .| + |.cos .| is (PI / 2) * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by L9;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: |.sin .| + |.cos .| is (PI / 2) * (k + 1) -periodic ; :: thesis: S₁[k + 1]
B1: ( dom (|.sin .| + |.cos .|) = REAL & dom |.sin .| = REAL & dom |.cos .| = REAL )

proof

B2: dom (|.sin .| + |.cos .|) = (dom |.sin .|) /\ (dom |.cos .|) by VALUED_1:def 1;
( dom |.sin .| = REAL & dom |.cos .| = REAL ) by SIN_COS:27, VALUED_1:def 11;
hence ( dom (|.sin .| + |.cos .|) = REAL & dom |.sin .| = REAL & dom |.cos .| = REAL ) by B2; :: thesis: verum

end;

|.sin .| + |.cos .| is (PI / 2) * ((k + 1) + 1) -periodic

proof

for x being real number st x in dom (|.sin .| + |.cos .|) holds
( x + ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & x - ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & (|.sin .| + |.cos .|) . x = (|.sin .| + |.cos .|) . (x + ((PI / 2) * ((k + 1) + 1))) )

proof

let x be real number ; :: thesis: ( x in dom (|.sin .| + |.cos .|) implies ( x + ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & x - ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & (|.sin .| + |.cos .|) . x = (|.sin .| + |.cos .|) . (x + ((PI / 2) * ((k + 1) + 1))) ) )
assume Z1: x in dom (|.sin .| + |.cos .|) ; :: thesis: ( x + ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & x - ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & (|.sin .| + |.cos .|) . x = (|.sin .| + |.cos .|) . (x + ((PI / 2) * ((k + 1) + 1))) )
(|.sin .| + |.cos .|) . (x + ((PI / 2) * ((k + 1) + 1))) = (|.sin .| . (x + ((PI / 2) * ((k + 1) + 1)))) + (|.cos .| . (x + ((PI / 2) * ((k + 1) + 1)))) by B1, VALUED_1:def 1
.= |.(sin . (x + ((PI / 2) * ((k + 1) + 1)))).| + (|.cos .| . (x + ((PI / 2) * ((k + 1) + 1)))) by VALUED_1:def 11, B1
.= |.(sin . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| + |.(cos . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| by VALUED_1:def 11, B1
.= |.(cos . (x + ((PI / 2) * (k + 1)))).| + |.(cos . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| by SIN_COS:83
.= |.(cos . (x + ((PI / 2) * (k + 1)))).| + |.(- (sin . (x + ((PI / 2) * (k + 1))))).| by SIN_COS:83
.= |.(cos . (x + ((PI / 2) * (k + 1)))).| + |.(sin . (x + ((PI / 2) * (k + 1)))).| by COMPLEX1:138
.= (|.cos .| . (x + ((PI / 2) * (k + 1)))) + |.(sin . (x + ((PI / 2) * (k + 1)))).| by VALUED_1:def 11, B1
.= (|.cos .| . (x + ((PI / 2) * (k + 1)))) + (|.sin .| . (x + ((PI / 2) * (k + 1)))) by VALUED_1:def 11, B1
.= (|.sin .| + |.cos .|) . (x + ((PI / 2) * (k + 1))) by VALUED_1:def 1, B1 ;
hence ( x + ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & x - ((PI / 2) * ((k + 1) + 1)) in dom (|.sin .| + |.cos .|) & (|.sin .| + |.cos .|) . x = (|.sin .| + |.cos .|) . (x + ((PI / 2) * ((k + 1) + 1))) ) by A3, Th1, B1, Z1; :: thesis: verum

end;

hence |.sin .| + |.cos .| is (PI / 2) * ((k + 1) + 1) -periodic by Th1; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
hence |.sin .| + |.cos .| is (PI / 2) * (k + 1) -periodic ; :: thesis: verum