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 Lm18;
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]
A4: ( dom (|.sin.| + |.cos.|) = REAL & dom |.sin.| = REAL & dom |.cos.| = REAL )

proof

A5: dom (|.sin.| + |.cos.|) = (dom |.sin.|) /\ (dom |.cos.|) by VALUED_1:def 1;
( dom |.sin.| = REAL & dom |.cos.| = REAL ) by SIN_COS:24, VALUED_1:def 11;
hence ( dom (|.sin.| + |.cos.|) = REAL & dom |.sin.| = REAL & dom |.cos.| = REAL ) by A5; :: thesis: verum

end;

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

proof

for x being Real 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; :: 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 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 A4, VALUED_1:def 1, XREAL_0:def 1
.= |.(sin . (x + ((PI / 2) * ((k + 1) + 1)))).| + (|.cos.| . (x + ((PI / 2) * ((k + 1) + 1)))) by A4, VALUED_1:def 11, XREAL_0:def 1
.= |.(sin . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| + |.(cos . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| by A4, VALUED_1:def 11, XREAL_0:def 1
.= |.(cos . (x + ((PI / 2) * (k + 1)))).| + |.(cos . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| by SIN_COS:78
.= |.(cos . (x + ((PI / 2) * (k + 1)))).| + |.(- (sin . (x + ((PI / 2) * (k + 1))))).| by SIN_COS:78
.= |.(cos . (x + ((PI / 2) * (k + 1)))).| + |.(sin . (x + ((PI / 2) * (k + 1)))).| by COMPLEX1:52
.= (|.cos.| . (x + ((PI / 2) * (k + 1)))) + |.(sin . (x + ((PI / 2) * (k + 1)))).| by A4, VALUED_1:def 11, XREAL_0:def 1
.= (|.cos.| . (x + ((PI / 2) * (k + 1)))) + (|.sin.| . (x + ((PI / 2) * (k + 1)))) by A4, VALUED_1:def 11, XREAL_0:def 1
.= (|.sin.| + |.cos.|) . (x + ((PI / 2) * (k + 1))) by A4, VALUED_1:def 1, XREAL_0:def 1 ;
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, A4, XREAL_0:def 1; :: 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