let k be Nat; |.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;
( S1[k] implies S1[k + 1] )
assume A3:
|.sin.| + |.cos.| is
(PI / 2) * (k + 1) -periodic
;
S1[k + 1]
A4:
(
dom (|.sin.| + |.cos.|) = REAL &
dom |.sin.| = REAL &
dom |.cos.| = REAL )
|.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;
( 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.|)
;
( 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;
verum
end;
hence
|.sin.| + |.cos.| is
(PI / 2) * ((k + 1) + 1) -periodic
by Th1;
verum
end;
hence
S1[
k + 1]
;
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
; verum