let k be Nat; :: thesis: cos is (2 * PI) * (k + 1) -periodic
defpred S1[ Nat] means cos is (2 * PI) * (\$1 + 1) -periodic ;
A1: S1[ 0 ] by Lm3;
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: cos is (2 * PI) * (k + 1) -periodic ; :: thesis: S1[k + 1]
cos is (2 * PI) * ((k + 1) + 1) -periodic
proof
for x being Real st x in dom cos holds
cos . x = cos . (x + ((2 * PI) * ((k + 1) + 1)))
proof
let x be Real; :: thesis: ( x in dom cos implies cos . x = cos . (x + ((2 * PI) * ((k + 1) + 1))) )
assume A4: x in dom cos ; :: thesis: cos . x = cos . (x + ((2 * PI) * ((k + 1) + 1)))
cos . (x + ((2 * PI) * (k + 1))) = cos . ((x + ((2 * PI) * (k + 1))) + (2 * PI)) by SIN_COS:78;
hence cos . x = cos . (x + ((2 * PI) * ((k + 1) + 1))) by A3, A4; :: thesis: verum
end;
then ( (2 * PI) * ((k + 1) + 1) <> 0 & ( for x being Real st x in dom cos holds
( x + ((2 * PI) * ((k + 1) + 1)) in dom cos & x - ((2 * PI) * ((k + 1) + 1)) in dom cos & cos . x = cos . (x + ((2 * PI) * ((k + 1) + 1))) ) ) ) by ;
hence cos is (2 * PI) * ((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 cos is (2 * PI) * (k + 1) -periodic ; :: thesis: verum