let a be Real; :: thesis: for x being FinSequence of REAL
for A being Matrix of REAL st len A = len x & len x > 0 & width A > 0 holds
(a * x) * A = a * (x * A)
let x be FinSequence of REAL ; :: thesis: for A being Matrix of REAL st len A = len x & len x > 0 & width A > 0 holds
(a * x) * A = a * (x * A)
let A be Matrix of REAL ; :: thesis: ( len A = len x & len x > 0 & width A > 0 implies (a * x) * A = a * (x * A) )
assume that
A1: len A = len x and
A2: len x > 0 and
A3: width A > 0 ; :: thesis: (a * x) * A = a * (x * A)
A4: (A @ ) * x = x * A by A1, A2, A3, Th52;
A5: width (A @ ) = len x by A1, A3, MATRIX_2:12;
then A6: (A @ ) * (a * x) = a * ((A @ ) * x) by A2, Th59;
A7: len (a * x) = len x by RVSUM_1:147;
len (A @ ) > 0 by A3, MATRIX_2:12;
then (a * x) * ((A @ ) @ ) = (A @ ) * (a * x) by A2, A5, A7, Th53;
hence (a * x) * A = a * (x * A) by A1, A2, A3, A6, A4, MATRIX_2:15; :: thesis: verum