for x being Real st x in dom () holds
( x + (PI / 2) in dom () & x - (PI / 2) in dom () & () . x = () . (x + (PI / 2)) )
proof
let x be Real; :: thesis: ( x in dom () implies ( x + (PI / 2) in dom () & x - (PI / 2) in dom () & () . x = () . (x + (PI / 2)) ) )
assume A1: x in dom () ; :: thesis: ( x + (PI / 2) in dom () & x - (PI / 2) in dom () & () . x = () . (x + (PI / 2)) )
A2: ( dom () = REAL & dom = REAL & dom = REAL )
proof end;
then () . (x + (PI / 2)) = ( . (x + (PI / 2))) + ( . (x + (PI / 2))) by
.= |.(sin . (x + (PI / 2))).| + ( . (x + (PI / 2))) by
.= |.(sin . (x + (PI / 2))).| + |.(cos . (x + (PI / 2))).| by
.= |.(cos . x).| + |.(cos . (x + (PI / 2))).| by SIN_COS:78
.= |.(cos . x).| + |.(- (sin . x)).| by SIN_COS:78
.= |.(cos . x).| + |.(sin . x).| by COMPLEX1:52
.= () + |.(sin . x).| by
.= () + () by
.= () . x by ;
hence ( x + (PI / 2) in dom () & x - (PI / 2) in dom () & () . x = () . (x + (PI / 2)) ) by ; :: thesis: verum
end;
hence |.sin.| + is PI / 2 -periodic by Th1; :: thesis: verum