let K be Field; :: thesis: for a, b being Element of K
for p, q being FinSequence of K st p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p = len q holds
(a * ()) + (b * ()) = SCirc ((a * p) + (b * q))

let a, b be Element of K; :: thesis: for p, q being FinSequence of K st p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p = len q holds
(a * ()) + (b * ()) = SCirc ((a * p) + (b * q))

let p, q be FinSequence of K; :: thesis: ( p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p = len q implies (a * ()) + (b * ()) = SCirc ((a * p) + (b * q)) )
set n = len p;
assume that
A1: p is first-symmetry-of-circulant and
A2: q is first-symmetry-of-circulant and
A3: len p = len q ; :: thesis: (a * ()) + (b * ()) = SCirc ((a * p) + (b * q))
A4: ( a * p is first-symmetry-of-circulant & b * q is first-symmetry-of-circulant ) by A1, A2, Th12;
A5: len (b * q) = len p by ;
(a * ()) + (b * ()) = (SCirc (a * p)) + (b * ()) by
.= (SCirc (a * p)) + (SCirc (b * q)) by
.= SCirc ((a * p) + (b * q)) by ;
hence (a * ()) + (b * ()) = SCirc ((a * p) + (b * q)) ; :: thesis: verum