let k be Nat; :: thesis: + is (PI / 2) * (k + 1) -periodic
defpred S1[ Nat] means + 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.| + is (PI / 2) * (k + 1) -periodic ; :: thesis: S1[k + 1]
A4: ( dom () = REAL & dom = REAL & dom = REAL )
proof end;
|.sin.| + is (PI / 2) * ((k + 1) + 1) -periodic
proof
for x being Real st x in dom () holds
( x + ((PI / 2) * ((k + 1) + 1)) in dom () & x - ((PI / 2) * ((k + 1) + 1)) in dom () & () . x = () . (x + ((PI / 2) * ((k + 1) + 1))) )
proof
let x be Real; :: thesis: ( x in dom () implies ( x + ((PI / 2) * ((k + 1) + 1)) in dom () & x - ((PI / 2) * ((k + 1) + 1)) in dom () & () . x = () . (x + ((PI / 2) * ((k + 1) + 1))) ) )
assume x in dom () ; :: thesis: ( x + ((PI / 2) * ((k + 1) + 1)) in dom () & x - ((PI / 2) * ((k + 1) + 1)) in dom () & () . x = () . (x + ((PI / 2) * ((k + 1) + 1))) )
() . (x + ((PI / 2) * ((k + 1) + 1))) = ( . (x + ((PI / 2) * ((k + 1) + 1)))) + ( . (x + ((PI / 2) * ((k + 1) + 1)))) by
.= |.(sin . (x + ((PI / 2) * ((k + 1) + 1)))).| + ( . (x + ((PI / 2) * ((k + 1) + 1)))) by
.= |.(sin . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| + |.(cos . ((x + ((PI / 2) * (k + 1))) + (PI / 2))).| by
.= |.(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
.= ( . (x + ((PI / 2) * (k + 1)))) + |.(sin . (x + ((PI / 2) * (k + 1)))).| by
.= ( . (x + ((PI / 2) * (k + 1)))) + ( . (x + ((PI / 2) * (k + 1)))) by
.= () . (x + ((PI / 2) * (k + 1))) by ;
hence ( x + ((PI / 2) * ((k + 1) + 1)) in dom () & x - ((PI / 2) * ((k + 1) + 1)) in dom () & () . x = () . (x + ((PI / 2) * ((k + 1) + 1))) ) by ; :: thesis: verum
end;
hence |.sin.| + 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.| + is (PI / 2) * (k + 1) -periodic ; :: thesis: verum