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 ;
then ( x + ((2 * PI) * (k + 1)) in () \ () & x - ((2 * PI) * (k + 1)) in () \ () ) by RFUNCT_1:def 2;
then ( x + ((2 * PI) * (k + 1)) in dom cos & not x + ((2 * PI) * (k + 1)) in cos " & x - ((2 * PI) * (k + 1)) in dom cos & not x - ((2 * PI) * (k + 1)) in cos " ) by XBOOLE_0:def 5;
then A5: ( not cos . (x + ((2 * PI) * (k + 1))) in & not cos . (x - ((2 * PI) * (k + 1))) in ) 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 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 " ) by ;
then A6: x + ((2 * PI) * ((k + 1) + 1)) in () \ () by XBOOLE_0:def 5;
x - ((2 * PI) * ((k + 1) + 1)) in dom cos by ;
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 " ) by ;
then A7: x - ((2 * PI) * ((k + 1) + 1)) in () \ () 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 ;
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 ;
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 ; :: 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