let a be Real; :: thesis: for x being FinSequence of REAL
for A being Matrix of REAL st width A = len x & len x > 0 holds
A * (a * x) = a * (A * x)
let x be FinSequence of REAL ; :: thesis: for A being Matrix of REAL st width A = len x & len x > 0 holds
A * (a * x) = a * (A * x)
let A be Matrix of REAL; :: thesis: ( width A = len x & len x > 0 implies A * (a * x) = a * (A * x) )
assume that
A1: width A = len x and
A2: len x > 0 ; :: thesis: A * (a * x) = a * (A * x)
A3: len (ColVec2Mx x) = len x by A2, MATRIXR1:def 9;
width (ColVec2Mx x) = 1 by A2, MATRIXR1:def 9;
then A4: 1 <= width (A * (ColVec2Mx x)) by A1, A3, MATRIX_3:def 4;
thus A * (a * x) = Col ((A * (a * (ColVec2Mx x))),1) by A2, MATRIXR1:47
.= Col ((a * (A * (ColVec2Mx x))),1) by A1, A3, MATRIXR1:40
.= a * (A * x) by A4, MATRIXR1:56 ; :: thesis: verum