let k be Nat; :: thesis: sec is (2 * PI) * (k + 1) -periodic
defpred S₁[ Nat] means sec is (2 * PI) * ($1 + 1) -periodic ;
A1: S₁[ 0 ] by Lm7;
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: sec is (2 * PI) * (k + 1) -periodic ; :: thesis: S₁[k + 1]
sec is (2 * PI) * ((k + 1) + 1) -periodic

proof

for x being Real st x in dom sec holds
( x + ((2 * PI) * ((k + 1) + 1)) in dom sec & x - ((2 * PI) * ((k + 1) + 1)) in dom sec & sec . x = sec . (x + ((2 * PI) * ((k + 1) + 1))) )

proof

let x be Real; :: thesis: ( x in dom sec implies ( x + ((2 * PI) * ((k + 1) + 1)) in dom sec & x - ((2 * PI) * ((k + 1) + 1)) in dom sec & sec . x = sec . (x + ((2 * PI) * ((k + 1) + 1))) ) )
assume x in dom sec ; :: thesis: ( x + ((2 * PI) * ((k + 1) + 1)) in dom sec & x - ((2 * PI) * ((k + 1) + 1)) in dom sec & sec . x = sec . (x + ((2 * PI) * ((k + 1) + 1))) )
then A4: ( x + ((2 * PI) * (k + 1)) in dom sec & x - ((2 * PI) * (k + 1)) in dom sec & sec . x = sec . (x + ((2 * PI) * (k + 1))) ) by A3, Th1;
then ( x + ((2 * PI) * (k + 1)) in (dom cos) \ (cos " {0}) & x - ((2 * PI) * (k + 1)) in (dom cos) \ (cos " {0}) ) by RFUNCT_1:def 2;
then ( x + ((2 * PI) * (k + 1)) in dom cos & not x + ((2 * PI) * (k + 1)) in cos " {0} & x - ((2 * PI) * (k + 1)) in dom cos & not x - ((2 * PI) * (k + 1)) in cos " {0} ) by XBOOLE_0:def 5;
then A5: ( not cos . (x + ((2 * PI) * (k + 1))) in {0} & not cos . (x - ((2 * PI) * (k + 1))) in {0} ) by FUNCT_1:def 7;
then cos . (x + ((2 * PI) * (k + 1))) <> 0 by TARSKI:def 1;
then cos . ((x + ((2 * PI) * (k + 1))) + (2 * PI)) <> 0 by SIN_COS:78;
then not cos . ((x + ((2 * PI) * (k + 1))) + (2 * PI)) in {0} by TARSKI:def 1;
then ( (x + ((2 * PI) * (k + 1))) + (2 * PI) in dom cos & not (x + ((2 * PI) * (k + 1))) + (2 * PI) in cos " {0} ) by FUNCT_1:def 7, SIN_COS:24, XREAL_0:def 1;
then A6: x + ((2 * PI) * ((k + 1) + 1)) in (dom cos) \ (cos " {0}) by XBOOLE_0:def 5;
x - ((2 * PI) * ((k + 1) + 1)) in dom cos by SIN_COS:24, XREAL_0:def 1;
then cos . (x - ((2 * PI) * ((k + 1) + 1))) = cos . ((x - ((2 * PI) * ((k + 1) + 1))) + (2 * PI)) by Lm3;
then ( x - ((2 * PI) * ((k + 1) + 1)) in dom cos & not x - ((2 * PI) * ((k + 1) + 1)) in cos " {0} ) by A5, FUNCT_1:def 7, SIN_COS:24, XREAL_0:def 1;
then A7: x - ((2 * PI) * ((k + 1) + 1)) in (dom cos) \ (cos " {0}) by XBOOLE_0:def 5;
then ( x + ((2 * PI) * ((k + 1) + 1)) in dom sec & x - ((2 * PI) * ((k + 1) + 1)) in dom sec ) by A6, RFUNCT_1:def 2;
then sec . (x + ((2 * PI) * ((k + 1) + 1))) = (cos . ((x + ((2 * PI) * (k + 1))) + (2 * PI))) " by RFUNCT_1:def 2
.= (cos . (x + ((2 * PI) * (k + 1)))) " by SIN_COS:78
.= sec . x by A4, RFUNCT_1:def 2 ;
hence ( x + ((2 * PI) * ((k + 1) + 1)) in dom sec & x - ((2 * PI) * ((k + 1) + 1)) in dom sec & sec . x = sec . (x + ((2 * PI) * ((k + 1) + 1))) ) by A6, A7, RFUNCT_1:def 2; :: thesis: verum

end;

hence sec 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 sec is (2 * PI) * (k + 1) -periodic ; :: thesis: verum