let k be Nat; :: thesis: sin (#) cos is PI * (k + 1) -periodic
defpred S₁[ Nat] means sin (#) cos is PI * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by Lm24;
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 * (k + 1) -periodic ; :: thesis: S₁[k + 1]
sin (#) cos is PI * ((k + 1) + 1) -periodic

proof

for x being Real st x in dom (sin (#) cos) holds
( x + (PI * ((k + 1) + 1)) in dom (sin (#) cos) & x - (PI * ((k + 1) + 1)) in dom (sin (#) cos) & (sin (#) cos) . x = (sin (#) cos) . (x + (PI * ((k + 1) + 1))) )

proof

let x be Real; :: thesis: ( x in dom (sin (#) cos) implies ( x + (PI * ((k + 1) + 1)) in dom (sin (#) cos) & x - (PI * ((k + 1) + 1)) in dom (sin (#) cos) & (sin (#) cos) . x = (sin (#) cos) . (x + (PI * ((k + 1) + 1))) ) )
assume A4: x in dom (sin (#) cos) ; :: thesis: ( x + (PI * ((k + 1) + 1)) in dom (sin (#) cos) & x - (PI * ((k + 1) + 1)) in dom (sin (#) cos) & (sin (#) cos) . x = (sin (#) cos) . (x + (PI * ((k + 1) + 1))) )
then A5: ( x + (PI * (k + 1)) in dom (sin (#) cos) & x - (PI * (k + 1)) in dom (sin (#) cos) ) by A3, Th1;
A6: ( x + (PI * ((k + 1) + 1)) in (dom cos) /\ (dom sin) & x - (PI * ((k + 1) + 1)) in (dom cos) /\ (dom sin) ) by SIN_COS:24, XREAL_0:def 1;
then ( x + (PI * ((k + 1) + 1)) in dom (sin (#) cos) & x - (PI * ((k + 1) + 1)) in dom (sin (#) cos) ) by VALUED_1:def 4;
then (sin (#) cos) . (x + (PI * ((k + 1) + 1))) = (sin . ((x + (PI * (k + 1))) + PI)) * (cos . ((x + (PI * (k + 1))) + PI)) by VALUED_1:def 4
.= (- (sin . (x + (PI * (k + 1))))) * (cos . ((x + (PI * (k + 1))) + PI)) by SIN_COS:78
.= (- (sin . (x + (PI * (k + 1))))) * (- (cos . (x + (PI * (k + 1))))) by SIN_COS:78
.= (sin . (x + (PI * (k + 1)))) * (cos . (x + (PI * (k + 1))))
.= (sin (#) cos) . (x + (PI * (k + 1))) by A5, VALUED_1:def 4 ;
hence ( x + (PI * ((k + 1) + 1)) in dom (sin (#) cos) & x - (PI * ((k + 1) + 1)) in dom (sin (#) cos) & (sin (#) cos) . x = (sin (#) cos) . (x + (PI * ((k + 1) + 1))) ) by A3, A4, A6, VALUED_1:def 4; :: thesis: verum

end;

hence sin (#) cos is 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 sin (#) cos is PI * (k + 1) -periodic ; :: thesis: verum