let x be FinSequence of REAL ; :: thesis: for A being Matrix of REAL st len x = width A & len A > 0 & len x > 0 holds
(- A) * x = - (A * x)
let A be Matrix of REAL; :: thesis: ( len x = width A & len A > 0 & len x > 0 implies (- A) * x = - (A * x) )
assume that
A1: len x = width A and
A2: len A > 0 and
A3: len x > 0 ; :: thesis: (- A) * x = - (A * x)
A4: ( len (ColVec2Mx x) = len x & width (ColVec2Mx x) = 1 ) by A3, MATRIXR1:def 9;
then A5: 1 <= width (A * (ColVec2Mx x)) by A1, MATRIX_3:def 4;
thus (- A) * x = Col ((((- 1) * A) * (ColVec2Mx x)),1) by Th9
.= Col (((- 1) * (A * (ColVec2Mx x))),1) by A1, A2, A3, A4, MATRIXR1:41
.= - (A * x) by A5, MATRIXR1:56 ; :: thesis: verum