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 L4;
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 number 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 number ; :: 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 A9: ( 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 8;
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 B2: ( not cos . (x + ((2 * PI ) * (k + 1))) in {0 } & not cos . (x - ((2 * PI ) * (k + 1))) in {0 } ) by FUNCT_1:def 13;
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:83;
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 13, SIN_COS:27;
then B3: x + ((2 * PI ) * ((k + 1) + 1)) in (dom cos ) \ (cos " {0 }) by XBOOLE_0:def 5;
cos . (x - ((2 * PI ) * ((k + 1) + 1))) = cos . ((x - ((2 * PI ) * ((k + 1) + 1))) + (2 * PI )) by L2, Th1, SIN_COS:27;
then ( x - ((2 * PI ) * ((k + 1) + 1)) in dom cos & not x - ((2 * PI ) * ((k + 1) + 1)) in cos " {0 } ) by B2, FUNCT_1:def 13, SIN_COS:27;
then B4: 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 B3, RFUNCT_1:def 8;
then sec . (x + ((2 * PI ) * ((k + 1) + 1))) = (cos . ((x + ((2 * PI ) * (k + 1))) + (2 * PI ))) " by RFUNCT_1:def 8
.= (cos . (x + ((2 * PI ) * (k + 1)))) " by SIN_COS:83
.= sec . x by A9, RFUNCT_1:def 8 ;
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 B3, RFUNCT_1:def 8, B4; :: 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