let k be Nat; :: thesis: cosec is (2 * PI) * (k + 1) -periodic
defpred S1[ Nat] means cosec is (2 * PI) * ($1 + 1) -periodic ;
A1: S1[ 0 ] by Lm6;
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: cosec is (2 * PI) * (k + 1) -periodic ; :: thesis: S1[k + 1]
cosec is (2 * PI) * ((k + 1) + 1) -periodic
proof
for x being Real st x in dom cosec holds
( x + ((2 * PI) * ((k + 1) + 1)) in dom cosec & x - ((2 * PI) * ((k + 1) + 1)) in dom cosec & cosec . x = cosec . (x + ((2 * PI) * ((k + 1) + 1))) )
proof
let x be Real; :: thesis: ( x in dom cosec implies ( x + ((2 * PI) * ((k + 1) + 1)) in dom cosec & x - ((2 * PI) * ((k + 1) + 1)) in dom cosec & cosec . x = cosec . (x + ((2 * PI) * ((k + 1) + 1))) ) )
assume x in dom cosec ; :: thesis: ( x + ((2 * PI) * ((k + 1) + 1)) in dom cosec & x - ((2 * PI) * ((k + 1) + 1)) in dom cosec & cosec . x = cosec . (x + ((2 * PI) * ((k + 1) + 1))) )
then A4: ( x + ((2 * PI) * (k + 1)) in dom cosec & x - ((2 * PI) * (k + 1)) in dom cosec & cosec . x = cosec . (x + ((2 * PI) * (k + 1))) ) by A3, Th1;
then ( x + ((2 * PI) * (k + 1)) in (dom sin) \ (sin " {0}) & x - ((2 * PI) * (k + 1)) in (dom sin) \ (sin " {0}) ) by RFUNCT_1:def 2;
then ( x + ((2 * PI) * (k + 1)) in dom sin & not x + ((2 * PI) * (k + 1)) in sin " {0} & x - ((2 * PI) * (k + 1)) in dom sin & not x - ((2 * PI) * (k + 1)) in sin " {0} ) by XBOOLE_0:def 5;
then A5: ( not sin . (x + ((2 * PI) * (k + 1))) in {0} & not sin . (x - ((2 * PI) * (k + 1))) in {0} ) by FUNCT_1:def 7;
then sin . (x + ((2 * PI) * (k + 1))) <> 0 by TARSKI:def 1;
then sin . ((x + ((2 * PI) * (k + 1))) + (2 * PI)) <> 0 by SIN_COS:78;
then not sin . ((x + ((2 * PI) * (k + 1))) + (2 * PI)) in {0} by TARSKI:def 1;
then ( (x + ((2 * PI) * (k + 1))) + (2 * PI) in dom sin & not (x + ((2 * PI) * (k + 1))) + (2 * PI) in sin " {0} ) by FUNCT_1:def 7, SIN_COS:24, XREAL_0:def 1;
then A6: x + ((2 * PI) * ((k + 1) + 1)) in (dom sin) \ (sin " {0}) by XBOOLE_0:def 5;
x - ((2 * PI) * ((k + 1) + 1)) in dom sin by SIN_COS:24, XREAL_0:def 1;
then sin . (x - ((2 * PI) * ((k + 1) + 1))) = sin . ((x - ((2 * PI) * ((k + 1) + 1))) + (2 * PI)) by Lm2;
then ( x - ((2 * PI) * ((k + 1) + 1)) in dom sin & not x - ((2 * PI) * ((k + 1) + 1)) in sin " {0} ) by A5, FUNCT_1:def 7, SIN_COS:24, XREAL_0:def 1;
then A7: x - ((2 * PI) * ((k + 1) + 1)) in (dom sin) \ (sin " {0}) by XBOOLE_0:def 5;
then ( x + ((2 * PI) * ((k + 1) + 1)) in dom cosec & x - ((2 * PI) * ((k + 1) + 1)) in dom cosec ) by A6, RFUNCT_1:def 2;
then cosec . (x + ((2 * PI) * ((k + 1) + 1))) = (sin . ((x + ((2 * PI) * (k + 1))) + (2 * PI))) " by RFUNCT_1:def 2
.= (sin . (x + ((2 * PI) * (k + 1)))) " by SIN_COS:78
.= cosec . x by A4, RFUNCT_1:def 2 ;
hence ( x + ((2 * PI) * ((k + 1) + 1)) in dom cosec & x - ((2 * PI) * ((k + 1) + 1)) in dom cosec & cosec . x = cosec . (x + ((2 * PI) * ((k + 1) + 1))) ) by A6, A7, RFUNCT_1:def 2; :: thesis: verum
end;
hence cosec 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 cosec is (2 * PI) * (k + 1) -periodic ; :: thesis: verum